I do not believe you are correct on the designation for a commercial flight if the president were on board. There is 'Executive One' if he flies on a US civil aircraft, but that is still federally owned.
> On December 26, 1973, to "set an example for the rest of the nation during the current energy crisis" and to "demonstrate his confidence in the airlines", then-President Richard Nixon became the only sitting president to travel on a regularly scheduled commercial airline flight when he flew on a United Airlines DC-10 from Washington Dulles International Airport to Los Angeles International Airport.[2] A Nixon aide carried a suitcase-sized secure communication device on board the plane, so that the President could remain in contact with Washington in the event of an emergency.