If its what I'm thinking of, it used to be a user-visible thing[1] back in the day.

It used to be user-visible, yes, but I wonder if TFA isn’t a little out-of-date, as the UI flow that used to work in order to see this (settings/icloud/keychain/advanced) isn’t there anymore on Mac or iOS. And random poking around indicates that they didn’t move it.

When one would be prompted to create a new code, the dialog said something about “changes to the servers” or something similar. Now, having read TFA, I wonder if that doesn’t mean an HSM got compromised somehow.