It's been a while since I was over my friends house and played the game I was talking about (it was one of those WWII shooters, the one that had the zombie minigame..) so I might not be 100% sure of this, but I remember havign to press the stick in AND push forward; in other words, pressing it in once wasn't like an "on" switch for run, it had to be held down the way you hold B down to run in SMB. Very annoying.
The first game I ever played that required a click of the control stick was Ape Escape on ps1. That was SUPER annoying because the stick was so flimsy that pressing it in made me push the stick in any given direction which made my character perform some other actions, perhaps movement? I can't remember it was so long ago.