Presidents who did not serve have no business saluting, but that’s my opinion.
I think I remember something being drilled into us about ‘without cover’ and ‘out of uniform’ as part of my training. Been a long time ago so I might be remembering incorrectly. President Reagan asked his Commanding General if he could salute and the CG told him he was President and could do whatever he wished, or something along those lines. Thus the Presidential salute was born and has been followed since.
That has actually been changed. I can't give you a source right now, but check with your local VA. But it was changed because of pressure from VA groups who felt that Vets should be able to salute the flag even though they are "out of uniform".