We've already seen that Presidents of the past flew on different planes. The first time the Presidential plane was referred to as Air Force One was during the Kennedy administration. Previous Presidents simply named their planes themselves.
Roosevelt, for reasons best known to himself, called his Douglas VC-54 C plane 'The Sacred Cow,' and had a special elevator fitted to it for his wheelchair. President Truman called his Douglas-built aircraft 'Independence,' after the Missouri town he grew up in.