Future of `DataArray.rename`

What is your issue?

In https://github.com/pydata/xarray/pull/6665 the question came up what to do with DataArray.rename in light of the new index refactor.

To be consistent with Dataset we should introduce a

• DataArray.rename_dims
• DataArray.rename_vars
• DataArray.rename

Several open questions about the behavior (Similar things apply to Dataset.rename{, _dims, _vars}):

• Should rename_dims also rename indexes (dimension coordinates)?
• Should rename_vars also rename the DataArray?
• What to do if the DataArray has the same name as one of its coordinates?
• Should rename still rename everything (like it is now) or only the name (Possibly with some deprecation cycle)?

The current implementation of DataArray.rename is a bit inconsistent:

As stated by @max-sixty in https://github.com/pydata/xarray/issues/6665#issuecomment-1154368202_:

• rename operates on DataArray as described in https://github.com/pydata/xarray/pull/6665#issuecomment-1150810485.%C2%A0Generally I'm less keen on "different types have different semantics", and here a positional arg would mean a DataArray rename, and kwarg would mean var rename. But it does work locally to DataArray quite well.
• rename only exists on DataArrays for the name of the DataArray, and we use rename_vars & rename_dims for both DataArrays & Datasets. So Dataset.rename is soft-deprecated.

Start by reading the current DataArray.rename implementation and the discussion in pull request 6665 and issue 6665. Review the corresponding Dataset.rename, rename_dims, and rename_vars behavior, then resolve the listed questions about indexes, variable names, name collisions, and deprecation before defining completion.