"RPG" isn't just some ubiquitous word we throw around to mean anything we want like "free love" ("If it's fun just do it!"). It has a distinct meaning, changing that meaning to "FUN!" or something else changes the word. People sitting around eating pizza and drinking beer while improving elves doesn't mean they're playing D&D even if it's a good time.
Hrm. A role-playing game is a game in which the (or a) primary purpose of the game is to undertake the role of one or more characters within the game milieu, and to make decisions from the perspective of the character(s) so undertaken. Because of this (1) rules that are dissociative (and thereby force the player to make choices from outside the stance of the characters) and (2) rules or set-ups that are railroady (and thereby force players to make decisions that the characters would not make, in some cases quite literally being forced to reverse decisions made from the character's stance because the GM does not like the outcome on "his story") damage the degree to which any game is a role-playing game.
But I am not about to say that, when Gary Gygax ran Steading of the Hill Giant Chief at GenCon, that he was not running a role-playing game, or that the players were not playing D&D.
2. "NPC" is only a word just like "dungeon" or "castle" or "monster". They are just props in a world. They don't create a story or a narrative on their own. They are only there because the players are interacting with them even if the DM controls their responses.
They are there because the dictates of the fictional milieu require them to be there. I am not engaging in games, like some computer games, where "NPCs" only exist to interact with the PCs. In my games, NPCs have their own plans and motives, introduce their own threads, and change the milieu thereby.
Because I am attempting to simulate a "breathing world", I don't want the PCs to be the only, or in some cases not even the primary, movers for all that happens. The world is not a vacuum, existing merely to cater to their whims. I would find such a set-up boring at best.
If the DM decides to write a back-story for his game and decides to tell his players about it, there is no RPG narrative happening. There is one-sided narrative, but unless the players are driving that narrative into something by their interaction, it is not an RPG.
So, if the PCs decide to seek rumours, while the GM is telling them what they learn, it is not an RPG? Sorry, but no.
Narrative driven by anything but the players is not an RPG. If they're not driving it, who is? The DM or a rule-book. Neither of which constitute the requirements for an RPG.
I call bullshit. Sorry, but no.
OC: "Sure!" says the player of the cleric character, "I'm moving over to the sacks now, sticking close to the lefthand wall."
DM: "Just as the three are about in position to look down the passages, and while the cleric is heading for the rotting bags, the magic-user cries out, and you see something black and nasty looking upon her shoulder!"
LC: "Hold on, Gary. Are you trying to drive the narrative here, by introducing some kind of monster!?! If you do that, this isn't a role-playing game!"
Absolutely not. And this is more than "given them (the players) some props" - it is setting the context and consequences of choices. And both context and perceived consequence drive the narrative as much as choices. They are only "some props" if the players get told "There is a spider here" and then get to decide what it does (setting the context) and what happens as a result (determining the consequence).
Role-playing game narrative is driven by the mutual interaction of the players and the game milieu as devised by the GM.
And, yes, an RPG run without NPCs and without a DM is absolutely possible. It may not be a table-top RPG like D&D that requires a DM. But it qualifies as an RPG nonetheless.
A "storytelling" game maybe, but not a role-playing game. You say,
The key is who is "driving" narrative? The players must do this and this alone for it to be an RPG.
I say, bullshit. The qualifying element for a role-playing game is a primary purpose of the players to undertake roles, and to make decisions within the framework of the game from the stance of those roles. Doing so requires context, and it requires consequence. It requires, in fact, a volleying of narrative control from player (narrative control over character's choices) to GM (narrative control over the context in which those choices occur, and the outcome/consequences thereof).
Without those elements of GM narrative control, there is no PC "stance" that has any meaning - the players are simply writing a collaborative story.
There are storytelling games. There are linear games. There are role-playing games. Too much loss of player agency creates a linear game. Too much loss of GM agency creates a storytelling game. The golden region between - where Player and GM agency volley and build off of each other - is where the role-playing game can be found.
And, certainly, that means that there is a border area where a game can be both a storytelling game and a role-playing game, or a linear game and a role-playing game. But, in neither case would I make the claim that the game was a good role-playing game.
In conclusion, I agree with GaelicVigil that "RPG" isn't just some ubiquitous word we throw around to mean anything we want" but I disagree entirely that it means what GaelicVigil seems to think it means. If it did, sitting around making up a story would be a "role-playing game", but playing D&D as described or played by Gary Gygax would not be.