In practice, yes. Pedantically, any Air Force aircraft carrying the president is Air Force One, no matter how it flies. Any Marine Corps aircraft will be Marine One. And as you might expect, there would be an Army One and Navy One as well.

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.