Marine One is the designation for any Marine Corps rotary winged aircraft that is carrying the President.
Small correction: Marine One is the designation for any Marine Corps aircraft that is carrying the President. Rotary wing or fixed wing.
So if the President hops a ride aboard a USMC F/A-18D or EA-6B ... or more likely (but still pretty unlikely) one of their Cessna Citation bizjet (see pic below), that aircraft bears the "Marine One" designation.
Technically, yes. However, Obama would have to get seat qualified to get a hop on a Delta Hornet - which the SS wouldn't let him fly aboard, or a Prowler and considering his physical frailties, that would be highly unlikely. I doubt he'd slum it on a UC-35 either.
Has any POTUS ever flown aboard a Marine fixed wing aircraft?