Michael said, "Nephew, call yourself Vincent Corleone." He didn't say, "call yourself Don Corleone," probably because Michael still regarded himself as the legitimate Don Corleone. Had Michael died and Vincent taken over, he probably would have called himself Don Corleone.
Carlo Gambino was known as "Don Carlo" by all. Castellano was known as "Paul," or "Paulie" by those reporting directly to him. I don't know what the lower-level people called him.