Do you simply mean regression testing?
- Aha, an expect test!
- Oh, you mean a snapshot test!
- This here is akin to UI testing framework X where the test framework can compare an expected screenshot of the UI to a screenshot of the actual UI!
The last one basically requires automation if you want anyone to make use of it. The regression testing automation described in the OP is a nice-to-have, not a so-good-that-it-gets-a-new-name.
"Regression test" means something else, at least at the companies I've worked at: It means a test that was written after a defect was found in production, to ensure that the same defect doesn't happen again (that the fix doesn't "regress"). It can be a manual test or an automated test. https://en.wikipedia.org/wiki/Regression_testing
One reason to call bug fix tests for “regression tests” (and only those kinds of tests) is that someone might regress the code base through a merge conflict (maybe they effectively undo a commit?). So that’s one argument I suppose.
I'm not particularly wedded to any of these terms, I'm just pointing out that "regression testing" has an established meaning, and it isn't snapshot testing (outside of certain industries, at least). I do find it amusing that one implementation of snapshot testing (https://pypi.org/project/pytest-regtest/) links to https://en.wikipedia.org/wiki/Regression_testing but that article doesn't describe snapshot testing at all! Maybe the article changed? Oh well, language changes too. ¯\_(ツ)_/¯