REGR: index_col False and header=None inferring index names in some cases
- [x] closes #46955 (Replace xxxx with the Github issue number)
- [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 an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.

The c engine raises as before.
thanks @phofl great as always @meeseeksdev backport 1.4.x Could not push to auto-backport-of-pr-47139-on-1.4.x due to error, aborting.