I prefer to roll and check without pinching at either edge, make adjustments and then pinch... in the center to ensure that it matches at both edges and then fold outward from the center. If you pinch at either edge and fold in you run the risk of the creases not matching perfectly at the center.
I understand why you like folding in thirds using that method, but I don't that it is ideal from a teaching perspective. The other method gives rise to a teachable moment: the introduction of Haga's first theorem. And the roll method or "s" method is applicable in only that one context, Haga's theorem allows for divisions into Nths and is thus more generalized and robust even if it might introduce extraneous pinching and folding. In my two week origami course I have not taught anything that required divisions by thirds but I do teach Haga's theorems.
