Let me give you a few examples.
You can use all-purpose flour to make a New York style pizza. That is what the early NYC pizza masters used before bread flour and, considerably later, high-gluten flour became more readily available. However, today, most NYC pizza operators use high-gluten flour. But, some, like many of us on the forum, use bread flour. There are still some operators who use all-purpose flour to make the NY style and would not think to use any other flour. Old habits die hard.
All-purpose flour is not the flour of choice for American style pizzas, such as those from Papa John's, Pizza Hut and Domino's. Their flours are likely to be proprietary flours milled to their specifications, but they are likely to be fairly high in protein, typically in the bread flour and high-gluten flour range.
All-purpose flour can also be used to make the Chicago deep-dish style although I believe that an all-purpose flour with a higher than normal protein content (like Ceresota or Hecker's) is quite common and maybe even preferred. A bread flour can also be used. You would not want to use high-gluten flour for this style but, no doubt, someone has tried it. And, maybe, even prefers it.
You can also use all-purpose flour to make a cracker style pizza. However, in my experience, the crust coloration is too light because of the low protein content of the all-purpose flour when compared with other flours. I personally like to use bread flour (my preference is the Better for Bread flour) but there are many who prefer using high-gluten flour. Of course, if you prefer a light colored crust for this style, then all-purpose flour can certainly be used.
Some people use all-purpose flour as a substitute for imported 00 flours to make the Neapolitan style pizza. However, I think that you will find that most people in the know would regard that as a poor substitute. The two flours are milled from different types of grains and do not produce the same results. Moreover, you really need a very high temperature oven to bake Neapolitan style pizzas. To approximate 00 flour, some people, especially those who do not have access to 00 flour, will use a mix of all-purpose flour and either cake flour or pastry flour. This was more common before 00 flours became readily available at the retail level. The flour combinations work to a point, but they are still not the same as 00 flours.
Possibly one of the better uses for all-purpose flour is for the California style where the star of the show is the exotic toppings more so than the crust, which should serve more as a canvas for the toppings.
You should also be able to use all-purpose flour to make the Sicilian style. But you can also use bread flour and some of the old timers in the field swear by high-gluten flour.
Most people tend to develop their preferences from using particular flours in specific recipes over a long period of time. Once the preferences are set, it is often hard to move off of them. That is why there are no right or wrong answers to your question. It usually comes down to choices and habits. If you look at dough recipes on this forum, you will see just about every flour has been used to make just about every type of pizza. The narrowest use of flour will be the 00 flour. It is used almost exclusively to make Neapolitan style pizzas. However, I once made a Neapolitan-inspired deep-dish pizza using 00 flour, and there are some who combine 00 flour with other flours, notably high-gluten flour, to make a New York Neapolitan version, such as available from DiFara's in Brooklyn where Dom DeMarco combines 00 flour with high-gluten flour.
As I noted before, all-purpose means all-purpose.