to_netcdf / open_dataset is not idempotent

[MVivien]

What happened:
I created a Dataset from a Dataarray with a data name equal to its dimension name and no coordinate. When saving the Dataset as netcdf and opening that netcdf as a Dataset again the opened Dataset does not have any data variable and the actual variable has become a coordinate.

What you expected to happen:
I would expect the to_netcdf / open_dataset process to be idempotent and obtain a Dataset that is identical to the one I saved as netcdf.

Minimal Complete Verifiable Example:

import xarray as xr

da = xr.DataArray(
    [1, 2, 3, 4],
    dims=['lat'],
    name='lat'
)
ds = da.to_dataset()

ds.to_netcdf('bug.nc')
ds2 = xr.open_dataset('bug.nc')

print(ds)
print(ds2)

Output

<xarray.Dataset>
Dimensions:  (lat: 4)
Dimensions without coordinates: lat
Data variables:
    lat      (lat) int64 1 2 3 4

<xarray.Dataset>
Dimensions:  (lat: 4)
Coordinates:
  * lat      (lat) int64 1 2 3 4
Data variables:
    *empty*

Anything else we need to know?:

Environment:

Output of xr.show_versions()

INSTALLED VERSIONS

commit: None
python: 3.6.12 |Anaconda, Inc.| (default, Sep 8 2020, 17:50:39)
[GCC Clang 10.0.0 ]
python-bits: 64
OS: Darwin
OS-release: 19.0.0
machine: x86_64
processor: i386
byteorder: little
LC_ALL: None
LANG: None
LOCALE: None.None
libhdf5: 1.10.6
libnetcdf: 4.7.4

xarray: 0.16.1
pandas: 1.1.3
numpy: 1.19.2
scipy: 1.5.2
netCDF4: 1.5.4
pydap: None
h5netcdf: None
h5py: None
Nio: None
zarr: None
cftime: 1.2.1
nc_time_axis: None
PseudoNetCDF: None
rasterio: None
cfgrib: 0.9.8.4
iris: None
bottleneck: None
dask: 2.30.0
distributed: None
matplotlib: 3.1.3
cartopy: None
seaborn: None
numbagg: None
pint: None
setuptools: 50.3.0.post20201006
pip: 20.2.3
conda: None
pytest: 6.1.0
IPython: 5.8.0
sphinx: None

[dcherian]

This is the same bug as in #4108 (comment)

<xarray.Dataset>
Dimensions:  (lat: 4)
Dimensions without coordinates: lat
Data variables:
    lat      (lat) int64 1 2 3 4

This isn't xarray's data model IIUC. Variables with the same name as dimensions are treated as coordinate variables (or indexed dimensions).