The relationship of presidents to their military is no different than that of a mayors/governors to their police departments: While they are in command of their military or police - the latter are not serving in the police and former are not serving in the military.
Don't believe me, look it up!
But according to the Constitution, the president is the commander-in-chief, and he has the ultimate authority over the armed forces. The National Museum of the Pacific War in Fredericksburg, Texas honors all of the presidents who served in the war and includes Truman and Roosevelt.