I would argue there’s a third Western style. Typically served round; thicker, more sour crust; a very generous cheddar/jack/mozzarella blend; and modest amounts of sauce, compared to Chicago-style anyway. The crust is often dusted with cornmeal. Round Table (founded in Menlo Park!) epitomized this pizza, though I haven’t eaten there in years so I can’t say if they’ve kept any semblance of quality.

You’ll find it throughout the Southwest at many smaller chains. Dion’s is one regional I can vouch for. Two Boots in New York is probably the closest I ever found on the East Coast, but I didn’t look hard outside of the City.

It’s distinct to me from New York style because of the amount of cheese and the rise of the crust. The more topping-laden variants (“combo”) can require a knife and fork to eat, whereas a classic New York slice is designed to be eaten standing up and toppings are portioned accordingly.