iCloud keeps asking for very old email (SOLVED!)

There's another discussion on this problem here. I'd love to be able to post a comment there to say that I found a solution that worked for me, but of course, it's closed now because that's how Apple's forum works. Sigh. So I have no choice but to start a brand new thread.

The problem: A prompt keeps popping up saying:

Sign in to iCloud

Enter the password for your Apple ID to continue using iCloud on this Mac

Below that, it has a password field for you to fill in, but it pre-fills the wrong Apple ID (email address) and doesn't let you change it! In our case, this was a very old email address that we originally used as our Apple ID but had changed many years ago.

The strangest thing about it is, the Mac was logged into iCloud already, using the correct Apple ID email address. The only service that we could not log into was FaceTime on the Mac.

So I called Apple Support and spent well over an hour discussing the issue with two different support people, and while they were very friendly and professional with lots of things to try, none of them worked.

During the call, I had an idea of my own. This old email address was not in any of our Apple ID settings, but it had to be coming from somewhere. What if it was coming from a saved password in Apple Keychain? So I opened the Keychain Access app, and searched for keychain tokens containing the old Apple ID. There were quite a number of them, but I couldn't delete them thanks to another bug with Keychain Access, discussed in this Ask Different thread. After the phone call, I went back and tried again, and eventually found and deleted the culprit. It turned out to be the one named 'com.apple.account.IdentityServices.token'. This one was linked to the old Apple ID account and after deleting it, the problems all went away. No more appearance of the annoying login prompt with the wrong Apple ID, and we are now able to login to FaceTime on the Mac.

I hope this helps someone else.