There are two specific planes they use. One bigger than the other depending on destination.

But no, those planes are only considered AF1 if the POTUS is a passenger. The designation follows the president.

Thats assuming a air force aircraft.

Marine aircraft is marine 1, specifically 1 of two helicopters usually used but as with AF1 it follows the potus to any marine aircraft.

Same goes with navy, aka navy 1. Or coast guard, which would be coast guard 1.

A private plane would be designated executive 1.

Anyhow. Yes af1 does include specific purpose built craft for POTUS but are only called such with the correct passenger.