No, it’s not true that any aircraft that The President is on receives the designation of Air Force One. It’s the designation of any Air Force airplane.
For instance, the helicopter that ferries him from The White House to Andrews Air Force Base is Marine One.
And the hot air balloon that Professor Moriarty uses to take The President off to his hot tub and Texas Hold ’em weekends is just a hot air balloon.
The President’s a lousy gambler, and he usually ends up handing over a lot of money.
But he can never hand over Air Force One.