I grew up in WNY and my Dad loved to make pizza from scratch ... the flour, yeast, kneading the dough and waiting for it to rise and then making a pizza. It was a Sunday night tradition. We'd eat the pizza holding it in our right hand, while holding a plate in our left hand under the pizza, to catch falling toppings. I doubt i ever had any pizzaria pizza until I was in my teens, and never even saw a square one until then, either.
These days I eat the corner with a knife and fork and perhaps hold it for the second half. However, I like a LOT of toppings on my pizza, so to avoid falling objects all over me, it's often left on the plate.
I'd never seen a folded in half pizza consumption until someone in a movie did it, and it looked rather strange. That usually doesn't work for thick pizzas or those with a lot of toppings.