That is where you start to get into semantics. Is a round pizza made with flatbread dough no longer a faltbread pizza because it is round? Is an hexagon pizza made with pizza dough no longer a pizza because it is neither round or square? Is a pizza without sauce no longer a pizza? Sometimes I think the question "What is the meaning of life?" is simpler to answer than "What is a pizza?".

For that reason I tend to go with things like ingredients as a more unambiguous classification method.
For me the dough is the first factor. A non-rising dough is a flatbread pizza. A rising dough is a pizza which can be further defined down to thin, hand tossed, pan, Chicago, Californian, etc.
Either way, if it tastes good you eat the sucker.