Refactor Dataset internals to store data variables and coordinate variables as separate dicts

There's a lot of other discussion in #9063, but I wanted to pull out this suggestion for independent discussion:

• Our list of internal attributes on a DataTree node is still not just those on a Dataset plus the inherited coordinates

This is indeed a bit of a con, but in my mind the right fix is probably to adjust the Dataset data model to using dictionaries of data_variables and coord_variables, rather than the current solution of a dict of variables and a set of coord_names. Using a separate dictionary for coord_variables would also be more aligned with how DataArray is implemented. The internal Dataset data model is a hold-over from the very early days of Xarray, before we had a notion of coordinate variables that are not indexes.

Originally posted by @shoyer in https://github.com/pydata/xarray/issues/9063#issuecomment-2198775511

Review the current Dataset model using a variables dictionary and coord_names, then compare it with DataArray's separate coordinate representation and the DataTree attribute concern described here. Identify the affected internal APIs and tests before changing them; done means Dataset stores data variables and coordinate variables separately while preserving the expected DataTree behavior.