To me, the defining characteristic of "American Pizza" is cutting any and all corners to reduce the need for skilled labor and to get the cost as low as possible.
Personally, I would not call NY-Style "American Pizza." Where there certainly is much similarity in the ingredients, toppings, assembly, etc., I'd argue there is almost no similarity in the philosophy. To the extent there is examples of "NY-Style" that fits into my definition above, I'd call them American Style, not NY-Style.