Air Force One is called that because it is operated by the Air Force. The President's personal helicopter is called Marine One because it is operated by the Marines.
What would the President's personal aircraft be called?
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I'd expect Executive One:
Executive One is the call sign designated for any United States civil aircraft when the President of the United States is on board.
Executive One is used while the President is on a civil aircraft according to FAA JO 7110.65W (.pdf). Apparently, this information hasn't been published in the AIM for a while.
(b) When the President is aboard a civil aircraft, state the words “Executive One.”