I decided to change my admin name. I went to "System Preferences"-> "Users & Groups" -> unlocked -> changed admin name.
After I rebooted my Mac, it seemed that Admin name hasn't changed. In Finder under "Users", I would still see only my old username. So I went into System Preferences again, but this time I would not be able to unlock.
Upon unlocking, it would show me a login menu WITHOUT username. Neither old nor new username works with the password (that remained unchanged). So basically I couldn't use my admin rights anymore.
Then I tried changing my password. Surprisingly, it seemed to work - the system accepted my old password and new password.
However I still cannot log in as admin. Whenever I press on the lock icon in Users & Groups, neither username/password combination works.