Chicago-style deep-dish pizza is closer to the former that you describe. From top to bottom on a "true" Chicago-style pizza, as made by Pizzeria Uno (the folks who invented it):
- Crushed tomatoes (no tomato sauce)
- Toppings
- Mozzerella cheese (the standard-sized Chicago-style, which is about 12" in diameter, apparently has a pound of cheese)
- Crust
I've lived in Chicago for 16 years, and have never heard anyone here refer to any pizza (either deep-dish or thin crust) as "pizza pie." Sounds East Coast-y to me, too.