diff --git a/dvc/command/diff.py b/dvc/command/diff.py
index 32ea8220d..650555572 100644
--- a/dvc/command/diff.py
+++ b/dvc/command/diff.py
@@ -95,7 +95,7 @@ class CmdDiff(CmdBase):
 
     def run(self):
         try:
-            diff = self.repo.diff(self.args.a_ref, self.args.b_ref)
+            diff = self.repo.diff(self.args.a_rev, self.args.b_rev)
 
             if not any(diff.values()):
                 return 0
@@ -131,20 +131,14 @@ def add_parser(subparsers, parent_parser):
         formatter_class=argparse.RawDescriptionHelpFormatter,
     )
     diff_parser.add_argument(
-        "a_ref",
-        help=(
-            "Git reference to the older version to compare "
-            "(defaults to HEAD)"
-        ),
+        "a_rev",
+        help="Old Git commit to compare (defaults to HEAD)",
         nargs="?",
         default="HEAD",
     )
     diff_parser.add_argument(
-        "b_ref",
-        help=(
-            "Git reference to the newer version to compare "
-            "(defaults to the current workspace)"
-        ),
+        "b_rev",
+        help=("New Git commit to compare (defaults to the current workspace)"),
         nargs="?",
     )
     diff_parser.add_argument(
diff --git a/dvc/command/metrics.py b/dvc/command/metrics.py
index 77d28bbf0..de11e3e9c 100644
--- a/dvc/command/metrics.py
+++ b/dvc/command/metrics.py
@@ -136,8 +136,8 @@ class CmdMetricsDiff(CmdBase):
     def run(self):
         try:
             diff = self.repo.metrics.diff(
-                a_ref=self.args.a_ref,
-                b_ref=self.args.b_ref,
+                a_rev=self.args.a_rev,
+                b_rev=self.args.b_rev,
                 targets=self.args.targets,
                 typ=self.args.type,
                 xpath=self.args.xpath,
@@ -283,20 +283,12 @@ def add_parser(subparsers, parent_parser):
         formatter_class=argparse.RawDescriptionHelpFormatter,
     )
     metrics_diff_parser.add_argument(
-        "a_ref",
-        nargs="?",
-        help=(
-            "Git reference to the older version to compare "
-            "(defaults to HEAD)"
-        ),
+        "a_rev", nargs="?", help="Old Git commit to compare (defaults to HEAD)"
     )
     metrics_diff_parser.add_argument(
-        "b_ref",
+        "b_rev",
         nargs="?",
-        help=(
-            "Git reference to the newer version to compare "
-            "(defaults to the current workspace)"
-        ),
+        help=("New Git commit to compare (defaults to the current workspace)"),
     )
     metrics_diff_parser.add_argument(
         "--targets",
diff --git a/dvc/exceptions.py b/dvc/exceptions.py
index 357f8c7cf..4af611108 100644
--- a/dvc/exceptions.py
+++ b/dvc/exceptions.py
@@ -112,18 +112,7 @@ class MoveNotDataSourceError(DvcException):
 
 
 class NotDvcRepoError(DvcException):
-    """Thrown if a directory is not a DVC repo.
-
-    Args:
-        root (str): path to the directory.
-    """
-
-    def __init__(self, root):
-        msg = (
-            "you are not inside of a DVC repository "
-            "(checked up to mount point '{}')"
-        )
-        super().__init__(msg.format(root))
+    """Thrown if a directory is not a DVC repo"""
 
 
 class DvcParserError(DvcException):
diff --git a/dvc/repo/__init__.py b/dvc/repo/__init__.py
index 9bdb04cd7..16062042f 100644
--- a/dvc/repo/__init__.py
+++ b/dvc/repo/__init__.py
@@ -129,19 +129,24 @@ class Repo(object):
 
     @classmethod
     def find_root(cls, root=None):
-        if root is None:
-            root = os.getcwd()
-        else:
-            root = os.path.abspath(os.path.realpath(root))
+        root_dir = os.path.realpath(root or os.curdir)
+
+        if not os.path.isdir(root_dir):
+            raise NotDvcRepoError("directory '{}' does not exist".format(root))
 
         while True:
-            dvc_dir = os.path.join(root, cls.DVC_DIR)
+            dvc_dir = os.path.join(root_dir, cls.DVC_DIR)
             if os.path.isdir(dvc_dir):
-                return root
-            if os.path.ismount(root):
+                return root_dir
+            if os.path.ismount(root_dir):
                 break
-            root = os.path.dirname(root)
-        raise NotDvcRepoError(root)
+            root_dir = os.path.dirname(root_dir)
+
+        message = (
+            "you are not inside of a DVC repository "
+            "(checked up to mount point '{}')"
+        ).format(root_dir)
+        raise NotDvcRepoError(message)
 
     @classmethod
     def find_dvc_dir(cls, root=None):
diff --git a/dvc/repo/diff.py b/dvc/repo/diff.py
index b8970c7b3..1f6dbeee6 100644
--- a/dvc/repo/diff.py
+++ b/dvc/repo/diff.py
@@ -6,7 +6,7 @@ from dvc.scm.git import Git
 
 
 @locked
-def diff(self, a_ref="HEAD", b_ref=None):
+def diff(self, a_rev="HEAD", b_rev=None):
     """
     By default, it compares the working tree with the last commit's tree.
 
@@ -47,8 +47,8 @@ def diff(self, a_ref="HEAD", b_ref=None):
         return result
 
     working_tree = self.tree
-    a_tree = self.scm.get_tree(a_ref)
-    b_tree = self.scm.get_tree(b_ref) if b_ref else working_tree
+    a_tree = self.scm.get_tree(a_rev)
+    b_tree = self.scm.get_tree(b_rev) if b_rev else working_tree
 
     try:
         self.tree = a_tree
diff --git a/dvc/repo/metrics/diff.py b/dvc/repo/metrics/diff.py
index e8b2151db..e3ad902d6 100644
--- a/dvc/repo/metrics/diff.py
+++ b/dvc/repo/metrics/diff.py
@@ -83,9 +83,9 @@ def _get_metrics(repo, *args, rev=None, **kwargs):
         return {}
 
 
-def diff(repo, *args, a_ref=None, b_ref=None, **kwargs):
-    old = _get_metrics(repo, *args, **kwargs, rev=(a_ref or "HEAD"))
-    new = _get_metrics(repo, *args, **kwargs, rev=b_ref)
+def diff(repo, *args, a_rev=None, b_rev=None, **kwargs):
+    old = _get_metrics(repo, *args, **kwargs, rev=(a_rev or "HEAD"))
+    new = _get_metrics(repo, *args, **kwargs, rev=b_rev)
 
     paths = set(old.keys())
     paths.update(set(new.keys()))

term: replace (Git) "reference" for "revision" or "commit"
Revisions are slightly more accurate - https://git-scm.com/docs/revisions

> A revision parameter '<rev>' typically, but not necessarily, names a commit object.

References are the ones living in `.git/refs`: https://git-scm.com/book/en/v2/Git-Internals-Git-References

We use the term _references_ in `metrics` and `diff` commands, but both commands accept revisions, since `repo.brancher` accepts `revs`(not `refs`).

This change would also require updating the docs.
diff --git a/tests/func/test_metrics.py b/tests/func/test_metrics.py
index 42043dbd8..09f9e229a 100644
--- a/tests/func/test_metrics.py
+++ b/tests/func/test_metrics.py
@@ -890,7 +890,7 @@ def test_metrics_diff_raw(tmp_dir, scm, dvc):
     _gen("raw 2")
     _gen("raw 3")
 
-    assert dvc.metrics.diff(a_ref="HEAD~2") == {
+    assert dvc.metrics.diff(a_rev="HEAD~2") == {
         "metrics": {"": {"old": "raw 1", "new": "raw 3"}}
     }
 
@@ -916,7 +916,7 @@ def test_metrics_diff_json(tmp_dir, scm, dvc, xpath):
     if not xpath:
         expected["m.json"]["a.b.e"] = {"old": "1", "new": "3"}
 
-    assert expected == dvc.metrics.diff(a_ref="HEAD~2")
+    assert expected == dvc.metrics.diff(a_rev="HEAD~2")
 
 
 def test_metrics_diff_broken_json(tmp_dir, scm, dvc):
@@ -939,4 +939,4 @@ def test_metrics_diff_broken_json(tmp_dir, scm, dvc):
 
 def test_metrics_diff_no_metrics(tmp_dir, scm, dvc):
     tmp_dir.scm_gen({"foo": "foo"}, commit="add foo")
-    assert dvc.metrics.diff(a_ref="HEAD~1") == {}
+    assert dvc.metrics.diff(a_rev="HEAD~1") == {}
diff --git a/tests/unit/command/test_metrics.py b/tests/unit/command/test_metrics.py
index fb2394bd2..856d24b48 100644
--- a/tests/unit/command/test_metrics.py
+++ b/tests/unit/command/test_metrics.py
@@ -30,8 +30,8 @@ def test_metrics_diff(dvc, mocker):
     m.assert_called_once_with(
         cmd.repo,
         targets=["target1", "target2"],
-        a_ref="HEAD~10",
-        b_ref="HEAD~1",
+        a_rev="HEAD~10",
+        b_rev="HEAD~1",
         typ="json",
         xpath="x.path",
         recursive=True,

pip install -e .[all]
parse_log_pytest
pytest
pytest pytest-timeout pytest-cov pytest-xdist pytest-mock flaky mock xmltodict awscli google-compute-engine Pygments collective.checkdocs flake8 psutil flake8-docstrings pydocstyle jaraco.windows mock-ssh-server moto rangehttpserver
apt-get update apt-get install -y gcc
3.7
pytest --no-header -rA --tb=line --color=no -p no:cacheprovider -W ignore::DeprecationWarning
aliyun-python-sdk-core==2.16.0
aliyun-python-sdk-core-v3==2.13.33
aliyun-python-sdk-kms==2.16.5
appdirs==1.4.4
atpublic==3.1.2
attrs @ file:///croot/attrs_1668696182826/work
autocommand==2.2.2
awscli==1.31.13
azure-common==1.1.28
azure-storage-blob==2.1.0
azure-storage-common==2.1.0
bcrypt==4.2.1
boto==2.49.0
boto3==1.33.13
botocore==1.33.13
cachetools==4.2.4
certifi @ file:///croot/certifi_1671487769961/work/certifi
cffi==1.15.1
charset-normalizer==3.4.1
collective.checkdocs==0.2
colorama==0.4.4
configobj==5.0.9
coverage==7.2.7
crcmod==1.7
cryptography==44.0.2
decorator==5.1.1
distro==1.9.0
docutils==0.16
-e git+https://github.com/iterative/dvc.git@7821fa6866a492c0675248028039257d292a09ec#egg=dvc
execnet==2.0.2
flake8==5.0.4
flake8-docstrings==1.7.0
flaky==3.8.1
flatten-json==0.1.14
flit_core @ file:///opt/conda/conda-bld/flit-core_1644941570762/work/source/flit_core
flufl.lock==7.1.1
funcy==2.0
future==1.0.0
gitdb==4.0.12
gitdb2==4.0.2
GitPython==3.1.44
google-api-core==2.10.2
google-api-python-client==2.166.0
google-auth==1.35.0
google-auth-httplib2==0.2.0
google-cloud-core==1.7.3
google-cloud-storage==1.19.0
google-compute-engine==2.8.13
google-crc32c==1.5.0
google-resumable-media==2.7.2
googleapis-common-protos==1.69.2
grandalf==0.6
httplib2==0.22.0
humanize==4.6.0
idna==3.10
importlib-metadata==4.2.0
importlib-resources==5.12.0
inflect==3.0.2
iniconfig @ file:///home/linux1/recipes/ci/iniconfig_1610983019677/work
jaraco.classes==3.2.3
jaraco.collections==4.2.0
jaraco.context==4.3.0
jaraco.functools==3.7.0
jaraco.structures==2.1.0
jaraco.text==3.11.1
jaraco.ui==2.3.0
jaraco.windows==5.7.0
Jinja2==3.1.6
jmespath==0.10.0
jsonpath-ng==1.7.0
MarkupSafe==2.1.5
mccabe==0.7.0
mock==5.2.0
mock-ssh-server==0.9.1
more-itertools==9.1.0
moto==4.2.14
nanotime==0.5.2
networkx==2.3
numpy==1.21.6
oauth2client==4.1.3
oss2==2.6.1
packaging @ file:///croot/packaging_1671697413597/work
paramiko==3.5.1
path==16.6.0
pathspec==0.11.2
pluggy @ file:///tmp/build/80754af9/pluggy_1648042572264/work
ply==3.11
protobuf==4.24.4
psutil==7.0.0
py @ file:///opt/conda/conda-bld/py_1644396412707/work
pyarrow==0.15.1
pyasn1==0.5.1
pyasn1-modules==0.3.0
pycodestyle==2.9.1
pycparser==2.21
pycryptodome==3.22.0
pydantic==1.10.21
pydocstyle==6.3.0
pydot==2.0.0
PyDrive2==1.15.4
pyfastcopy==1.0.3
pyflakes==2.5.0
Pygments==2.17.2
PyNaCl==1.5.0
pyOpenSSL==25.0.0
pyparsing==3.1.4
pytest==7.1.2
pytest-cov==4.1.0
pytest-mock==3.11.1
pytest-timeout==2.3.1
pytest-xdist==3.5.0
python-dateutil==2.8.0
PyYAML==5.1.2
rangehttpserver==1.4.0
requests==2.31.0
responses==0.23.3
rsa==4.7.2
ruamel.yaml==0.18.10
ruamel.yaml.clib==0.2.8
s3transfer==0.8.2
shortuuid==1.0.13
six==1.17.0
smmap==5.0.2
snowballstemmer==2.2.0
speedcopy==2.1.5
texttable==1.7.0
tomli @ file:///opt/conda/conda-bld/tomli_1657175507142/work
tqdm==4.67.1
treelib==1.7.1
types-PyYAML==6.0.12.12
typing_extensions @ file:///croot/typing_extensions_1669924550328/work
uritemplate==4.1.1
urllib3==1.26.20
voluptuous==0.14.1
Werkzeug==2.2.3
xmltodict==0.14.2
zc.lockfile==3.0.post1
zipp @ file:///croot/zipp_1672387121353/work

name: dvc
channels:
  - defaults
  - https://repo.anaconda.com/pkgs/main
  - https://repo.anaconda.com/pkgs/r
  - conda-forge
dependencies:
  - _libgcc_mutex=0.1=main
  - _openmp_mutex=5.1=1_gnu
  - attrs=22.1.0=py37h06a4308_0
  - ca-certificates=2025.2.25=h06a4308_0
  - certifi=2022.12.7=py37h06a4308_0
  - flit-core=3.6.0=pyhd3eb1b0_0
  - importlib_metadata=4.11.3=hd3eb1b0_0
  - iniconfig=1.1.1=pyhd3eb1b0_0
  - ld_impl_linux-64=2.40=h12ee557_0
  - libffi=3.4.4=h6a678d5_1
  - libgcc-ng=11.2.0=h1234567_1
  - libgomp=11.2.0=h1234567_1
  - libstdcxx-ng=11.2.0=h1234567_1
  - ncurses=6.4=h6a678d5_0
  - openssl=1.1.1w=h7f8727e_0
  - packaging=22.0=py37h06a4308_0
  - pip=22.3.1=py37h06a4308_0
  - pluggy=1.0.0=py37h06a4308_1
  - py=1.11.0=pyhd3eb1b0_0
  - pytest=7.1.2=py37h06a4308_0
  - python=3.7.16=h7a1cb2a_0
  - readline=8.2=h5eee18b_0
  - setuptools=65.6.3=py37h06a4308_0
  - sqlite=3.45.3=h5eee18b_0
  - tk=8.6.14=h39e8969_0
  - tomli=2.0.1=py37h06a4308_0
  - typing_extensions=4.4.0=py37h06a4308_0
  - wheel=0.38.4=py37h06a4308_0
  - xz=5.6.4=h5eee18b_1
  - zipp=3.11.0=py37h06a4308_0
  - zlib=1.2.13=h5eee18b_1
  - pip:
      - aliyun-python-sdk-core==2.16.0
      - aliyun-python-sdk-core-v3==2.13.33
      - aliyun-python-sdk-kms==2.16.5
      - appdirs==1.4.4
      - atpublic==3.1.2
      - autocommand==2.2.2
      - awscli==1.31.13
      - azure-common==1.1.28
      - azure-storage-blob==2.1.0
      - azure-storage-common==2.1.0
      - bcrypt==4.2.1
      - boto==2.49.0
      - boto3==1.33.13
      - botocore==1.33.13
      - cachetools==4.2.4
      - cffi==1.15.1
      - charset-normalizer==3.4.1
      - collective-checkdocs==0.2
      - colorama==0.4.4
      - configobj==5.0.9
      - coverage==7.2.7
      - crcmod==1.7
      - cryptography==44.0.2
      - decorator==5.1.1
      - distro==1.9.0
      - docutils==0.16
      - dvc==0.83.0+7821fa
      - execnet==2.0.2
      - flake8==5.0.4
      - flake8-docstrings==1.7.0
      - flaky==3.8.1
      - flatten-json==0.1.14
      - flufl-lock==7.1.1
      - funcy==2.0
      - future==1.0.0
      - gitdb==4.0.12
      - gitdb2==4.0.2
      - gitpython==3.1.44
      - google-api-core==2.10.2
      - google-api-python-client==2.166.0
      - google-auth==1.35.0
      - google-auth-httplib2==0.2.0
      - google-cloud-core==1.7.3
      - google-cloud-storage==1.19.0
      - google-compute-engine==2.8.13
      - google-crc32c==1.5.0
      - google-resumable-media==2.7.2
      - googleapis-common-protos==1.69.2
      - grandalf==0.6
      - httplib2==0.22.0
      - humanize==4.6.0
      - idna==3.10
      - importlib-metadata==4.2.0
      - importlib-resources==5.12.0
      - inflect==3.0.2
      - jaraco-classes==3.2.3
      - jaraco-collections==4.2.0
      - jaraco-context==4.3.0
      - jaraco-functools==3.7.0
      - jaraco-structures==2.1.0
      - jaraco-text==3.11.1
      - jaraco-ui==2.3.0
      - jaraco-windows==5.7.0
      - jinja2==3.1.6
      - jmespath==0.10.0
      - jsonpath-ng==1.7.0
      - markupsafe==2.1.5
      - mccabe==0.7.0
      - mock==5.2.0
      - mock-ssh-server==0.9.1
      - more-itertools==9.1.0
      - moto==4.2.14
      - nanotime==0.5.2
      - networkx==2.3
      - numpy==1.21.6
      - oauth2client==4.1.3
      - oss2==2.6.1
      - paramiko==3.5.1
      - path==16.6.0
      - pathspec==0.11.2
      - ply==3.11
      - protobuf==4.24.4
      - psutil==7.0.0
      - pyarrow==0.15.1
      - pyasn1==0.5.1
      - pyasn1-modules==0.3.0
      - pycodestyle==2.9.1
      - pycparser==2.21
      - pycryptodome==3.22.0
      - pydantic==1.10.21
      - pydocstyle==6.3.0
      - pydot==2.0.0
      - pydrive2==1.15.4
      - pyfastcopy==1.0.3
      - pyflakes==2.5.0
      - pygments==2.17.2
      - pynacl==1.5.0
      - pyopenssl==25.0.0
      - pyparsing==3.1.4
      - pytest-cov==4.1.0
      - pytest-mock==3.11.1
      - pytest-timeout==2.3.1
      - pytest-xdist==3.5.0
      - python-dateutil==2.8.0
      - pyyaml==5.1.2
      - rangehttpserver==1.4.0
      - requests==2.31.0
      - responses==0.23.3
      - rsa==4.7.2
      - ruamel-yaml==0.18.10
      - ruamel-yaml-clib==0.2.8
      - s3transfer==0.8.2
      - shortuuid==1.0.13
      - six==1.17.0
      - smmap==5.0.2
      - snowballstemmer==2.2.0
      - speedcopy==2.1.5
      - texttable==1.7.0
      - tqdm==4.67.1
      - treelib==1.7.1
      - types-pyyaml==6.0.12.12
      - uritemplate==4.1.1
      - urllib3==1.26.20
      - voluptuous==0.14.1
      - werkzeug==2.2.3
      - xmltodict==0.14.2
      - zc-lockfile==3.0.post1
prefix: /opt/conda/envs/dvc

tests/func/test_metrics.py::test_metrics_diff_raw tests/func/test_metrics.py::test_metrics_diff_json[True] tests/func/test_metrics.py::test_metrics_diff_json[False] tests/func/test_metrics.py::test_metrics_diff_no_metrics tests/unit/command/test_metrics.py::test_metrics_diff
tests/func/test_metrics.py::TestMetricsCLI::test
tests/func/test_metrics.py::test_show_dirty tests/func/test_metrics.py::TestMetrics::test_formatted_output tests/func/test_metrics.py::TestMetrics::test_show tests/func/test_metrics.py::TestMetrics::test_show_all_should_be_current_dir_agnostic tests/func/test_metrics.py::TestMetrics::test_type_case_normalized tests/func/test_metrics.py::TestMetrics::test_unknown_type_ignored tests/func/test_metrics.py::TestMetrics::test_xpath_all tests/func/test_metrics.py::TestMetrics::test_xpath_all_columns tests/func/test_metrics.py::TestMetrics::test_xpath_all_rows tests/func/test_metrics.py::TestMetrics::test_xpath_all_with_header tests/func/test_metrics.py::TestMetrics::test_xpath_is_empty tests/func/test_metrics.py::TestMetrics::test_xpath_is_none tests/func/test_metrics.py::TestMetricsRecursive::test tests/func/test_metrics.py::TestMetricsReproCLI::test tests/func/test_metrics.py::TestMetricsReproCLI::test_binary tests/func/test_metrics.py::TestMetricsReproCLI::test_dir tests/func/test_metrics.py::TestMetricsCLI::test_binary tests/func/test_metrics.py::TestMetricsCLI::test_dir tests/func/test_metrics.py::TestMetricsCLI::test_non_existing tests/func/test_metrics.py::TestMetricsCLI::test_wrong_type_add tests/func/test_metrics.py::TestMetricsCLI::test_wrong_type_modify tests/func/test_metrics.py::TestMetricsCLI::test_wrong_type_show tests/func/test_metrics.py::TestNoMetrics::test tests/func/test_metrics.py::TestNoMetrics::test_cli tests/func/test_metrics.py::TestCachedMetrics::test_add tests/func/test_metrics.py::TestCachedMetrics::test_run tests/func/test_metrics.py::TestMetricsType::test_show tests/func/test_metrics.py::test_display_missing_metrics tests/func/test_metrics.py::test_show_xpath_should_override_stage_xpath tests/func/test_metrics.py::test_show_multiple_outputs tests/func/test_metrics.py::test_metrics_diff_broken_json tests/unit/command/test_metrics.py::test_metrics_show_json_diff tests/unit/command/test_metrics.py::test_metrics_show_raw_diff tests/unit/command/test_metrics.py::test_metrics_diff_no_diff tests/unit/command/test_metrics.py::test_metrics_diff_no_changes
dvc/command/diff.py dvc/exceptions.py dvc/command/metrics.py dvc/repo/metrics/diff.py dvc/repo/__init__.py dvc/repo/diff.py
dvc/command/diff.py dvc/exceptions.py dvc/command/metrics.py dvc/repo/metrics/diff.py dvc/repo/__init__.py dvc/repo/diff.py