Don’t take the names too seriously. Probably not really useful at all. Just a thought...out loud. Not really very developed.
Old school: It’s a mostly free-form game with some simple, abstract rules for whatever we think it would be fun to have rules for.
New school: It’s about the world, PC backgrounds, and story. We need rules for things other than combat and magic.
Deconstructionist school: Let’s deconstruct the hobby and rebuild it in other interesting ways. We take the free-form elements from the old school, the world/background/story focus from the new school, and make any rules even simpler and more abstract.
Forum school: Forget the rules and dice. We’re going 100% free-form.
Engineer school: Let’s try to make the rules to cover everything. Which we think is rejecting new school and harkening back to old school.