How can I not have Command Line Tools installed, yet still have it?
Git being there means some part of the CLT is there, since the shim doesn't come up, and since Xcode is not installed.
It is a known problem that the receipt for the CLT may disappear.  One documentation of this is by MacPorts, linked here.

Download the command line tools package from the ​Apple Developer site.

If the version of the CLT you need is not listed at the developer site, but an older version is, reinstalling that will restore the package receipts. Then just run Software Update to trigger the update to the newer version.

Or, on Mac OS X 10.9 or greater, Software Update can reinstall the package:
sudo touch /tmp/.com.apple.dt.CommandLineTools.installondemand.in-progress
softwareupdate -l


​Run Software Update
sudo rm /tmp/.com.apple.dt.CommandLineTools.installondemand.in-progress