Correct.
AF1 (Navy1, Marine1, Army1): Prez on board.
Executive1: Prez on board civilian aircraft.
AF1F/Executive1F, etc: Prez family on board aircraft but no Prez ("Executive one foxtrot" for instance).
AF2 (etc): Vice Prez.
Same aircraft without the Prez, VP, or their family: some regularly allocated callsign or the actual tailnumber. But NOT Air Force 1 (etc).
Dave