Would it count as a different "action" or "input" if pressing a button does something different from tapping it? Is it against the spirit of making the game have only one button? 

Yes, tap and hold is allowed.

It still counts as one button/input, because the player is using the same physical button.

So a quick tap can do one thing, and holding the same button can do something else.

Or tap-tap, tap-tap-tap… long hold, short hold, etc… I can think of quite a few “combos” (all with one button) that would work - depending on the type of game…