I'm sure they used the "badnik" name in the Sonic 3 manual. They had this section which described the different badniks in the game. They gave them names, too.
From Wikipedia,
"A Badnik is a term for an enemy robot in the early North American and European localizations of the Sonic the Hedgehog video games."
"Note: Starting with Sonic Adventure, the term "Badnik" fell out of use. In Sonic Adventure and all the subsequent games the robots are only referred to as "Eggman's robots". They will still be included in this article."