To me, this seems to be an issue of what we call 'prototypal classification' (sometimes called 'fuzzy classification' when it's computers doing the sorting). Basically, you have two groupings (pizza crust and cracker), and you have to decide how to sort a specific item (the crust on a thin crust pizza).
It's important to consider that crackers and pizza crusts are both rather wide categories, and what items we're most familiar with in that category will affect how we judge other items for inclusion.
Some examples of 'pizza' and pizza-like items:
- New York thin-crust pizza
- Chicago deep-dish pizza
- Pizza bianco
- Barbeque chicken pizza
- Dessert pizza with mascarpone & fruit
- French bread pizza
- Pizza bagels
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And some types of cracker and cracker like items:
- Oyster crackers
- Graham crackers
- Bagel chips
People may only be familiar with some of those, and some may not consider all of those items to be part of the category. (Zwieback is a rusk, and therefore, a type of biscotti ... does that make it a cookie? Although technically, biscotti is a twice cooked, so it's a 'biscuit', but the term 'biscuit' has diverged in American English).
This is part of why we get disagreements on what a 'pizza' is. If you grew up with New York thin-crust pizza, and only that style of pizza, it's difficult to consider Chicago deep-dish pizza to be a part of your mental image of 'pizza'. It's closer to a casserole or a fruit buckle than a New York style pizza with its crisp crust and slightly charred top.
Likewise, to someone who's grown up with deep dish pizzas, a New York style pizza is missing the breadiness and chew of the pizza you're familiar with, and it's closer to a cracker with some sauce and cheese on top.
As for where the two diverge -- that's even more difficult to say, because it's entirely possible that we could find some cases where there's agreement that an item is in both categories.
If we can agree that tomato isn't integral to pizza, then a crust topped with olive oil, garlic and cheese is still a pizza. If the type of cheese doesn't matter, then we can use a hard grating cheese like parmesean or pecorino romano.
And if we can agree that crackers don't have to be bite sized, and to matzo is a cracker, and that crackers can be flavored (quite common now from Wheat Thins and Triscuits in the stores) then should you use those ingredients on a New York style crust, you would have something that would be exceedingly difficult to classify as either a pizza or a cracker and not both.