Which is really my Apple ID password?

I just changed the password for my Mom's Apple devices with her iPhone. When I switched to her iPad, it recognized the password it had was invalid and I entered the new one and it was accepted. Then on her Mac Mini I went to apple.com and tried to log in, but it would not accept the new password. It accepted an old one, perhaps coincidentally, perhaps not, the Mac Mini system password. I have tried to use Safari -> Preferences -> Passwords to keep track of the password without success. It was set to an old password and I was not allowed to update it or change it where it took me to change it. The web page for this has a big Apple ID label in the upper-left, implying that that password is for my Apple ID. The passwords are labeled likewise on the iPhone and iPad, but the password is different. Apparently Apple doesn't want to document how this all works on pages where understanding them is needed, so can someone explain all the different passwords involved?

Additionally, every few weeks when a device demands her Apple ID password, the one we set previously never works. I always have to change it again to be able to log back in to the device. What is doing on with this? I noticed the keychain is turned on on the iPhone and iPad. I'm not sure what this has to do with, since I can no longer trust my understanding of what should be obvious (how a keychain should operate).