This classification system is deeply flawed but one of the most obvious ways is failing to recognize that quiche is an arbitrarily over specific example of what its category should ACTUALLY be called, which is obviously PIE.

PIZZA IS PIE TOO. The crust puffing up elevated at the edges contains the ingredients within.

And in this case, a stuffed crust pizza is indeed a PIE SURROUNDED BY A CALZONE.

Alternatively surrounded by a burrito.