MuseScore 4 is not reading existing parts from version 3 properly

I have searched for this issue and haven't found any useful answer.
Problem description:
If I just double click on a score written in MS3 it will automatically open in MS4. MS4 will open it with an asterisk on the name tab, which i have interpreted as "this is not the correct version, you have to save it for it to be able to work properly". And I did that. MS4 shows only the main score, not the parts I had already created in MS3.
If I go to File->Parts to create (or open?) them, MS4 opens all my parts from MS3 and double them again with the original (midi?) names from MS4. That means, if I have e.g. a well formatted Tenor Sax Score in MS3 (which I renamed "Tenor" back then), MS4 will open it twice, once as "Tenor" and once as "Tenor Saxophone". If I go to Edit->Instruments i can only see "mine" (the renamed, well-formated ones from MS3) but if I go to File->Parts i have a looooong list of instruments (20 parts, my band consists of 10 musicians).
Closing the "bad" tabs is not really helpful, because they remain listed not only in File->Parts but also in File->Export.
Am I doing something wrong?
My Operative System is MacOS Ventura on an iMac and MacOS Monterey on a MacBook Air. Both of them with MS3 and MS4 installed
Thanks a lot for your time :-) and a happy 2023