I set a hint, it normally helps, but not in this case. I don't know why.
Besides, that's somehow not the answer to the question of my last reply. That is: is there anywhere a backup on the computer (e.g. somewhere in the ~/Library folder) of the old keychain before I switched (just enabled that, please see screenshot above) the user from Standard to Admin?
Once more: I have keychain enabled, everything worked fine, until I flipped that switch "Allow this user to administer this computer" to ON. This requires a reboot and after the reboot the login and password stored in keychain were gone.
It's nice to have this community help system. But why I always get the strange feeling, the people answering do not carefully read the question? Answering a lot of questions helps to collect points, doesn't it? Just asking.
The other question was "under which name is the password for the TimeMachine disk stored?" - prior to finding it as you suggest, I should at least get a hint what I should be looking for.
As far as I see it, switching a user to become an admin is - typical for Unix OSs - only adding it to a group having admin rights. That should by no way, never ever, affect the keychain. Specially without any hint or warning. I expect to have exactly the same situation It's a bug, not a feature ➡️ For your convenience: I'm Unix sysadmin (several Linuxes, Solaris, AIX).