Will the real Apple ID stand up?

I just changed the password for my (Mom's) Apple ID with her iPhone. When I used her iPad, it recognized the password it had was invalid and I entered the new one and it was accepted. On her Mac Mini I went to apple.com and tried to log in and it required a different password—coincidentally or not—the Mac Mini system password. The web page that deals with this has a big "Apple ID" in the upper left. The password stored in Safari Preferences for apple.com was neither of these and an attempt to change it "at the web site" didn't work. Can someone explain the different passwords involved? I see that "Use Keychain" is turned on on the iPhone and iPad. Also, I would appreciate an explanation of why that every few weeks when something demands her Apple ID password (such as having to relog in to a device), it never accepts the password we set and I have to change it again to be able to log in.

All devices are using the latest MacOS, I think. The above problems, where it makes sense, also existed in the previous version.