I honestly don’t think that whether you’re compressing the crust or not is a worthwhile test for cutting pizza. It’s something that matters when you’re slicing a loaf of bread or a cake, but pizza is pizza, and there’s a reason why the lower part of it is called a pizza crust rather than a pizza loaf.
All four of the methods I can think of (using a round pizza cutter, using scissors, drawing a small knife across it, and chopping it with a large knife) would be good enough, as long as all the blades are sharp. It almost seems to me that when it comes to perfecting pizza-cutting, most people would get more benefit from perfecting the surface it’s sitting on. Almost everyone has a cutter or knife that’s good enough for pizza (though some of them may need sharpening); lots of people don’t have an extra-large cutting board that would make the job more practical and easier on their knives. The only other useful specialized pizza-cutting tool I’ve seen is a chopping knife that’s long enough to comfortably chop an extra-large pizza in half at a single stroke; most people would rightly hesitate to keep a knife that big in their kitchen, but in a pizza restaurant it certainly moves things along.
Maybe that just turned into an accidental vote for scissors? 
ETA: I just remembered that my parents (though they didn’t make pizza) had a strange extra-large cutting board that got pulled out when needed - it was just a fairly big piece that had been left over when they installed their countertops. It would be a disaster in a butcher shop, but a makeshift board would certainly help if you were going to be cutting a lot of pizza and didn’t want to waste money on a Special Super Expensive Specialized Pizza Cutting Surface. (Plus you didn’t want your guests hacking holes into your counters, or beating your knives into plowshares.) Where you cut pizza is just going to get all scratched up anyway, right? Maybe the Pizza Steel will now meet its rightful companion the Pizza ScrapBoard.
