Kelly, what you're describing, from what I can ascertain, is pretty much a picture perfect NY Style pizza. Crispy from the oven, but foldable after sitting, golden, puffy, thin, but not cracker-y, large bubbles, gummy layer of sauce when the cheese is removed. I'm not sure that 'rubbery' is the best adjective to use for the cooled cheese, though. If the cheese hasn't melted quite enough and bubbled, then it might actually be a bit rubbery, but properly bubbled mozzarella isn't rubbery, but more, for lack of a better word, firm or chewy. You basically end up with a web of firm cheese interwoven with milkfat. It's almost like the kind of framework you see in a high fat hamburger, but, instead of brown strands of meat surrounded by pockets of fat, you've got milky white strands of cheese. There's no stringiness to it when you pull the slices apart, but the cooled cheese has a very complex string-ish structure inside.
I went to college in the St. Louis area from 85 to 89. I highly doubt this is the same place, but I distinctly recall St. Louis having at least one half decent NY style pizzeria. No idea what the name was- all I can recall was that the exterior and interior had a ton of red brick and that the pizza was good- not as good as what I grew up with back in NY, but still good. If St. Louis could have one good NY style pizzeria then, they could certainly have had more.