Just to add a new twist to this issue (and maybe some useful information). Although the problem seems to be linked with the cloud, with the Music app ignoring changes done in the Mac and overriding them with what was on the cloud, it is also present offline.
As mentioned before, I've made the decision to try to downgrade to BigSur. In the meantime, as I just can't stay without listening music while working, I decided tu turn off the wi-fi in my Mac while listening to music. Also I figured, now that I'm offline I might as well continue rating the songs I have not rated and tweaking the playlists as I want, then do a full time machine backup, downgrade and then in Big Sur every change will be there. So I rated around 50 songs while offline and added a few to a playlist. Then I had to turn on the wi-fi for something else, so before doing that I shut down the Music App. I worked online for a while, then turned off the wi-fi, and opened the Music App again. What do you think happened?
Yes, of the 50 songs I had manipulated offline, around 30 lost the rating and a few (not all) changes I have done in playlists were gone. This was without any synch to the cloud, which tells me the problem does lie within Monterey 12.1
Naively, I tried to see an upside of this and thought "well at least not ALL the songs lost the rating and SOME of the playlist changes were maintained, that's more than what I have achieved lately, so at least sync this to the cloud and that would be something" Yeah right... so I turned on the wi-fi, synched to the cloud, and then ALL changes were lost and my library reverted to the way it was like a week ago haha. (well, ratings and playlists changes, because I've noticed if I buy new songs, those remain).
I so wish Apple fixes this soon....