You are all right in a way. There are two aspects to this.
1) What most people call Air Force One is a specific plane (think there are 2), outfitted for the president and his entourage, which we all know and love. This does NOT appear to be the plane that Bidan got on.
2) Any plane the president rides on should be call-signed as AF-1 by the Air Force and MLATS or whatever system. Bidan's flight to Delaware was NOT call-signed as such. Which is either a slap in his face by the Air Force, or an effort to obscure his whereabouts…..or refelctive of the fact that he is not the recognized president.