"Enter the password you use to unlock the Mac mini."

I'm having problems getting AppStore to show me updates (it accepts my Apple ID password, then throws an error at me), so I thought I'd log out of iCloud and log back in again to see if I could clear it.

So I sign back in to iCloud, and it asks me for my Apple ID password. I give it. Then it asks me for the Mac password of the user account. I give that. Then it throws this up:

What the h*l is that?

It doesn't like my account password, it doesn't like my Apple ID, and I don't have any more passwords to throw at it. I don't have a firmware password or any other special thing.

I finally got around it by using the "Forgot... password" link and supplying the unlock code for my iPhone... but seriously, what the heck is it asking for here, and how can I either find out the right answer or stop it from asking me this again?

Also, when it threw up the list of devices I could use codes from to complete the sign-in process, I saw a lot of wonky and long-gone devices that don't even show up in the devices list at icloud.com. How can I clear those out?