I think it is clear there's a bunch of people here who would like to 'take back' the usage that Edwards coined 20 years ago, with the additional caveat that he doesn't label anything as being 'narrative'. He DOES define, or at least use, the terms Narrativist and Narrativism as terms however, with a fairly specific meaning. While there may be some people who would rather his usage was forgotten, or feel that their own ideas better fit that label, that is unfortunately not how the world works, generally. Edwards' coinage was fair, he explains it in terms of NOT reusing other existing terms in a confusing way!
Okay. I'm glad you got that off your chest
In any case, 'general methods of resolution' is pretty vague, all games have 'methods of resolution' and I would consider most of them to be pretty 'general'.
Most RPGs rate things in terms of success, or failure, possibly with degrees. Some have complications attached to certain actions. In PbtA games, the best games, and the best rules in those games, have a pretty specific result when certain conditions occur. They are discrete results. They are "general" in the sense of being versatile, and up to you to color with specific images or details, but they give you something.
As opposed to a game telling me, "Now the character suffers in escalation in Despair." I am in exactly the same situation as when I'm staring at a blank page of my novel, and I think, "Now I think the character suffers from despair at their plight. What causes this and what form does it take?" I know from my experience doing free-form, that ideally, you don't have too many of these moments. They are a lot of work. Sometimes they are too artificial and deflate pathos.
I don't understand what you even mean by this, how does play proceed from here?
You tell me. I was informed above that some games have "hope" as the possible outcome of a check.
In my view, every resolution in a game should GIVE you something. GURPS or D&D gives you success or failure. PbtA, the same (at least, the better ones). A good storytelling-oriented game gives you seeds and things to act on, and then encourages you to let those things unfold.
A game that says, "Now come up with an exciting way the character succeeds," is ASKING for something. A game that is just a series of storytelling prompts is a game that feeds on human blood. Ideally, a very mechanistic game, as I would define it, gives clear outcomes with enough room to describe the result. Ideally, a good narrative game provides a general outcome, and frames it in such a way that the group is empowered to follow the consequences to their logical ends, and spawn new situations to evaluate. It's a problem when a mechanistic game occasionally tells you to just "go wild." It's a problem when a narrative game doesn't give you enough framing to actually resolve anything.
Games that are at more of a midpoint between mechanistic and narrative can have either or both sets of those kinds of problems. My big complaint about Dungeon World is that it does things I don't want it to, and doesn't do things I do; it constrains a lot of choices by defining very specific things about characters and certain procedures, but then some of the Moves just ask me to make up stuff on the spot. So it's challenging to "re-skin" the mechanistic elements, and challenging to get the "make story" parts of the game to answer questions raised by various tests and challenges. So it has a little bit of leakage at both ends.
An example of a very narrative game I was in was a forum-based Vampire game. It used the Vampire: The Masquerade rules for chargen but was played mostly diceless. The main rules were that you couldn't do more than two "scenes" per day in your posts, you had to acknowledge the passage of time implied by other people's scenes, and you couldn't kill, permanently imprison, or "out" anyone's character as anything without their permission. Within those precepts, someone could just post one day, "Jason is out for his evening prowl, looking for possible victims. He's in the club district. He looks dressed for a night out, but his dark attire seems a little out of place." So any player can then respond, and anyone who wants to can establish they are taking actions that same night. That's narrative. Virtually everything is handled by first narrating it, then resolving any conflicts between actions. All the players and all the mods have a great deal of authority to narrative The World. It's very different from a more mechanistic game that treats each player character, each vehicle, as a conveyance from the player's senses, with some authority like the GM telling them what they experience. People advance the game literally just by telling stories, even narrating things outside their character's control; this is different from a mechanistic game, where their success is likely dependent on their personal competence, and their attempt to perform tasks over which they have control.