FYI for Americans: when you describe pizza as pie and non-American English speakers are listening, their inner child recoils in horror.
pizza = flat thing with cheese on top, pizza base at bottom
pie = closed thing with significant vertical component, pastry on top and bottom
Otherwise, every flatbread or similar thing on earth is a pie. Observe: "Oh, I really dig these Indian roti-pies". "Pass me another naan-pie". "Papadam pie party!". "The omlette is my favourite pie!". "Cheese toasties are a way better pies than pizza!". "A breakfast of pancake: how I love a pie in the morning!" It's insane.
First of all, a pie must be baked, so that would eliminate omelettes, pancakes, and more from your list. Second, a pie must have a crust and a filling. That eliminates a lot of the list too.
Second, a pie needn't have pastry on both top and bottom. There are numerous pies with no crust on top, such as pumpkin pie, key lime pie, banana cream pie, lemon meringue pie, chess pie, pecan pie, or chocolate icebox pie. These are so similar to pies with crust on top that it is hard to see how a reasonable definition could include one but not the other.
Toasted sandwiches are baked and by your definition could still then be pie. If you would argue otherwise, where do you draw the line between baking and toasting?
While the crust doesn't have to be on both sides, there does still need to be some kind of crust. So that disqualifies lasagna and moussaka, at least in any form I've ever had them. It also probably disqualifies most if not all casseroles, since they don't have a crust, or at least I wouldn't call it that. (In my experience, many casseroles have no breading. But, in those that do, it's usually either mixed in with everything or is lightly sprinkled on top.)
I wouldn't consider a toasted sandwich to have a crust because to me a crust is more of a hard pastry layer, not a fluffy piece of bread which has been browned.
Though that is a bit of a gray area. If you were to take some fluffy white bread, get it wet, and mash it down into a layer that lines a pie pan, and then put some filling (fruit for example) in it and bake it, and if the bread all hardened together into a cohesive outer shell, then I suppose I'd call that a very unconventional pie.
Just to make everything more complicated, there are deep-dish pizzas that have a very pie-like form. There's crust on top and the bottom, and they are cooked in a pan that resembles a pie pan. For example, this: http://winespiritsbeer.blogspot.com/2013/09/beer-crust-deep-... . Compared to a flat pizza, I think those are more deserving of the word "pie" not only because there is a crust but because the ingredients are more of a filling and less of a topping.
Another question is whether a (baked) calzone or empanada is a pie. They certainly meet most of my criteria, so maybe I'd have to say they are.
No, casserole and pie are different. Casseroles are typically multiple layers and/or a mixture of many ingredients in one pan. Pies have a single, distinct pastry/dough base that contains a filling.
Don't all non-Americans (like me), whenever they see pie in this context, quietly sing to themselves, "When a moon hits your eye like a big pizza pie..."?
Something you may or may not be aware of: You can purchase or make a double-crust pizza; the upper crust can have ingredients as well. If you do it "deep dish" style, you have the pizza equivalent of a "pizza pie" - completely sealed.
This isn't quite the same as a calzone (which is usually a single serving affair - think of it as a large pizza-filled pasty).
Honestly, though - aside from getting it to cook properly - I'm not sure that a pie crust with pizza filling would be that bad; blind-bake the crust, add the fillings, cook at as hot as you can get your oven, maybe broil the cheese on top (or take a torch to it).
pizza = flat thing with cheese on top, pizza base at bottom
pie = closed thing with significant vertical component, pastry on top and bottom
Otherwise, every flatbread or similar thing on earth is a pie. Observe: "Oh, I really dig these Indian roti-pies". "Pass me another naan-pie". "Papadam pie party!". "The omlette is my favourite pie!". "Cheese toasties are a way better pies than pizza!". "A breakfast of pancake: how I love a pie in the morning!" It's insane.