BUG: Improve MultiIndex label rename checks, docs and tests
- [x] closes #55169 
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.
Actually, I think I need to write an additional test for when `errors="raise"` and a rename is successful against a `MultiIndex`. That fails without this patch because of the inconsistency between the check and transform logic  Hi @phofl , @mroeschke,
Would you be able to help review this PR? This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Thanks for the pull request, but it appears to have gone stale. If interested in continuing, please merge in the main branch, address any review comments and/or failing tests, and we can reopen. Hi @mroeschke,
I've updated my branch to resolve merge conflicts. The new tests are still passing on my machine.
Could you kindly reopen this PR? It appears I cannot reopen this PR as there was a force push. Would be happy to have a new PR with your changes