Yes, that is the normal experience outside the Midwest/Chicago. We’re special here. 
When I was growing up in the early 1970s we would have family parties at Marion’s Piazza in Dayton, Ohio. By default they cut the pizzas into small squares, and I think they were one of the first to do so in the area (if not the country). So that’s what I grew up with, and is what I prefer today. (I just did a little research, and it would appear they may have copied St. Louis–style pizza.)
It’s the standard cut for thin crust pizza here in Chicago. I can’t figure out who first came up with the “party cut”/“tavern cut”/“square cut” pizza, but it’s pretty common around parts of the Midwest. Like you noted, St. Louis does it, I see it all around the Milwaukee area and beyond in Wisconsin, NW Indiana does it; I’ve seen it as far east as Columbus. (Though now that I think of it, I feel that there are some joints in New Jersey that do a square cut as well.)
Oddly, I’ve been to Chicago, and eaten pizza there. But somehow ended up with only wedge-cut pizzas.
“Regular” is something similar to a pizza Margherita. It’s a fairly thin , tossed crust and the pizza is round. It’s usually baked right on the floor of the oven with no pan. ( or a baking stone at home) “Sicilian” is more like focaccia. It’s thicker crust, it isn’t tossed and the dough is spread out on a pan.
Interesting thank you. What about ‘regular’ pizza? I’m just interested on what someone in a certain place might regard as regular, as obviously pizza varies wildly.
Then you didn’t have our indigenous tavern style pizza. You probably had deep dish, which is not our day-to-day pizza.
Or you went to Chicago and found yourself at a Pizza Hut 
For next time in town:
Someone noted above that New York is the only place to get a decent Sicilian pizza. I haven’t been in the Philadelphia are recently but while here may be a good Sicilian style to be found here and there around the country even in Philadelphia where the Neapolitan style pizzas are as good as you’ll find anywhere (except maybe Italy) the Sicilian pizzas in my experience didn’t match those from New York. Not that just any New York pizzeria made them great either. I haven’t had a pizza from New York recently either but I found places that had a good Sicilian moved a lot of them. A single one was usually made from at least 1-1/2 ordinary pizza doughs, with a lot more sauce that would bake into the thicker crust.
One thing making them slightly less popular was the long time prep time, first baked with just sauce, a little more slowly than thinner pizzas to get more rise from the dough, in a couple of places I see them starting them in a ordinary oven then the cheese and toppings added for more cook time, and finished in a hotter pizza oven. So it wasn’t a pizza to get in a hurry, and the price was scaled for extra ingredients plus the additional prep time.
Some places offered slices for sale, I usually saw that in places doing a brisk lunch time business. I can’t get a decent Sicilian around here, none of the pizza around here is all that great and some of it clearly being purchased by people who have never had better.
I had both deep dish and “regular” pizza, at places recommended by the organizer of the event I went to. (who lives in Chi town.) But no, I didn’t go to any taverns.
They’re not just served at taverns (I would say the majority aren’t.)The ma and pop neighborhood joints make that style and cut it that way, too. If you find slices, it’s typically at places doing a non-Chicago style of pizza, like, Coalfire and Spacca Napoli and the such which do East Coast and Neapolitan style pies. Chicago has become quite diverse pizza wise since my childhood, when everything served at parties was what became to be known as tavern-style pizza. We didn’t call it that growing up. It was just pizza, and it was always served party cut, and default toppings were always sausage (and mushroom, green pepper, and onion if going deluxe.) not a pepperoni town.
Squares is a Minnesota thing - “fer sure!” In fact it’s sometimes referred to as Sota Style. Almost all of the pizza places here other than Pizza Hut and Little Caesars cut their pizzas in squares.
Triangles. The pointy end makes pizza easier to eat.
I prefer pizza with light sauce and extra cheese. I dislike tomatoes and can’t stand chunky sauce on my pizza. Any high acid food gives me GI problems.
There’s some chains I avoid because their pizza has a lot of sauce. I definitely avoid deep dish.
Whoa! I prefer heavy sauce, normal or less cheese.
And rectangular pizzas are also always large. Large is squares medium to small is triangles.
Probably wigged out at the mere thought of having to eat square pizza.
Triangles is the only way to go. Every piece has some crust in which to hold it. Inner squares are absolutely a mess to pickup & hold
I eat inner squares with a fork and knife.
Barbarian!
See, that’s just wrong. The piece that someone puts on your plate determines how you eat it???