Type 3 is about story - wanting to have a game with a satisfying narrative. A beginning, a middle and an end and have it all "make sense". That's the "N" in GNS.
The prior Threefold (GDS) model used that and called it Dramatism. Part of the whole kerfuffle over GNS Narrativism was the Story Now essay, in which Ron Edwards, to put it mildly, radically altered the frame of conversation around story/narrative/dramatic play. He put "satisfying narrative" (GDS Dramatism) under Type 2 (or "S"), which ruffled quite a few feathers—and still does!
So we can add a Type 4, which is about generating events full of pressure and propulsion - wanting to have a game that constantly puts you on the spot, shoves hard decisions in your face, and is always escalating something. This is the "Story Now" others have already mentioned, and it's quite a different creature. The name's misleading at best but let's not get into that. In fact, I'm not even going to get into Story Now more than that, but some particular things that folks developed as a result of that essay & discussion.
Yep, I'm talking about Apocalypse World and its many offshoots. I'm not going to get deep into theory either, just talk about a few things this branch of game evolution brought about that 5e doesn't really cater to:
One is that dice rolls are not yes/no. "Nothing happens"
never happens in Apocalypse World. You wouldn't be rolling the dice otherwise. Generally you're going to get what you want (or more!), get (some of) what you want but with complications, or things are gonna go pear-shaped on you, and hard. 5e could introduce this type of resolution in particular contexts, and has some optional rules for it (including marginal failures and crit/fumble rules variants), but a lot of the core game mechanics are premised on the spell just not taking effect, or the sword not connecting. Apocalypse World notably has no blow-by-blow rules for combat. Different priorities.
Another is that players usually know what the possible outcomes are before they roll the bones. The potential events are typically—not necessarily—presented in the open, and possibly even negotiated between GM and players (this negotiation process is quite prominent in Blades in the Dark). It's still possible for the GM to put forward something like, "If you miss, well, you don't know what'll happen, but you know you won't like it," of course. In any case, the
odds are out in the open, and part of the process of play in these games is negotiating those odds. Once the dice fall, everybody knows how things are gonna shake out.
Also, the typical process involves the generation of fiction (or possible fiction) in the moment, rather than an exploration of a pre-written state of affairs, which is more typical in 5e. Prior events inform the moment, of course, but the moment itself can go in all sorts of improvised directions. Here again, the systems support that more, or less. As one example, Apocalypse World and Blades in the Dark keep stats and such to a minimum, so that a GM can whip up an NPC or whatever on the fly. 5e is more heavyweight, so improv is harder (but not impossible).
That's just three things, there are more. And it isn't so much theory as specific design choices, but I figured it was worth getting into that in order to give something of an impression of what Story Now proposed.