According to Gorbachev himself, the US made no such promise. According to that interview, the commonly cited quote from Secretary of State James Baker, "NATO will not move one inch further east," is taken wildly out of context. It was made during talks over the reunification of Germany:

...making sure that NATO’s military structures would not advance and that additional armed forces from the alliance would not be deployed on the territory of the then-GDR after German reunification. Baker’s statement, mentioned in your question, was made in that context.

With regards to Germany, they were legally enshrined and are being observed.

He also said this, without further elaboration:

[Expanding NATO east] was definitely a violation of the spirit of the statements and assurances made to us in 1990.

Here is where I think Gorbachev's interview comes in for some legit criticism. I honest find this a bit perplexing. Putting severe limitations on NATO membership, knowing that many countries would want to join, was a big ask. The proper thing is to write that out in legal language, translated into Russian and English, and mutually agreed upon. This feels like the geopolitical version of empty "thoughts and prayers."