Save Transaction: Is the name local to the stored procedure or not?
No they aren't local to a stored procedure.

If you run the following in a database with simple recovery model

checkpoint
begin tran;
exec dbo.foo_test_tran_outer;
commit tran;
select * from dbo.footest;

select Operation,
       Context,
       [Savepoint Name],
       [Transaction Name] ,
       Description 
from sys.fn_dblog(NULL,NULL)


It gives these results

+--------------------+----------+-------------------+------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     Operation      | Context  |  Savepoint Name   | Transaction Name |                                                                                               Description                                                                                                |
+--------------------+----------+-------------------+------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| LOP_BEGIN_CKPT     | LCX_NULL | NULL              | NULL             |                                                                                                                                                                                                          |
| LOP_END_CKPT       | LCX_NULL | NULL              | NULL             | log_minRecoveryLsn 00000112:00001ce6:008e;log_replbeginlsn 00000000:00000000:0000;log_replnextlsn 00000000:00000000:0000;log_distbackuplsn 00000000:00000000:0000;log_distlastlsn 00000000:00000000:0000 |
| LOP_BEGIN_XACT     | LCX_NULL | NULL              | user_transaction | user_transaction;0x010500000000000515000000007dfcff481d5bed8a729370ee030000                                                                                                                              |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             |                                                                                                                                                                                                          |
| LOP_MARK_SAVEPOINT | LCX_NULL | the_constant_name | NULL             |                                                                                                                                                                                                          |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             |                                                                                                                                                                                                          |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             |                                                                                                                                                                                                          |
| LOP_MARK_SAVEPOINT | LCX_NULL | the_constant_name | NULL             |                                                                                                                                                                                                          |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             |                                                                                                                                                                                                          |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             | COMPENSATION                                                                                                                                                                                             |
| LOP_LOCK_XACT      | LCX_NULL | NULL              | NULL             | COMPENSATION                                                                                                                                                                                             |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             | COMPENSATION                                                                                                                                                                                             |
| LOP_MODIFY_ROW     | LCX_HEAP | NULL              | NULL             | COMPENSATION                                                                                                                                                                                             |
| LOP_LOCK_XACT      | LCX_NULL | NULL              | NULL             | COMPENSATION                                                                                                                                                                                             |
| LOP_COMMIT_XACT    | LCX_NULL | NULL              | NULL             |                                                                                                                                                                                                          |
+--------------------+----------+-------------------+------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


The [Savepoint Name] recorded in the transaction log does not contain any unique identifier related to stored procedure. 

If you remove the rollback tran the_constant_name; from foo_test_tran_inner then the result of the final select changes to Inner, before savepoint showing that the rollback tran the_constant_name; executed from foo_test_tran_outer just rolls back to the most recent (unrolled back) savepoint of that name and takes no account of stored procedure nesting.