The primary goal of a wargame is to fight, thus a wargame.
The primary goal of a roleplaying game is to roleplay, thus a roleplaying game.
The primary goal of a storytelling game is to create/tell stories.
This is creating a false dichotomy.
More than that: it's also making up terns and definitions.
I've heard of minis games that basically revolve around combat, sure. And in that case the role-playing is basically out just as it is in a game of RISK or Poker. But I haven't heard them called War-Games, and the primary goal isn't to fight, but to have fun and win against the other player through strategy.
I've heard of role-playing games, but they've run the gamut from WoW to Final Fantasy to D&D to T20 to GURPS to, heck, Magic the Gathering in some instances. Some of them have you playing against a computer or other players, others feature a GM to give you obstacles, but that you're not playing "against," in the competitive sense.
I've heard of RPG's that focus on narrative elements like plot, scenes, character development, etc., but these are not an entirely different species of game as much as they are a refinement of role-playing to a specific end. Mostly these are like other RPG's, but they have a special kind of focus.
I've also heard of games that are about storytelling: things like a "pass the baton" kind of game where one person writes a sentence, the second person writes the next sentence, etc. These games don't concern themselves with playing a role, but they are about telling a story. They don't go around identifying as "storytelling games" as far as I know, they're just "games." Games in which you happen to tell a story, like Hungry Hungry Hippos is a game in which you happen to slam a lever as fast as possible (which actually skirts kind of close to an RPG what with your little plastic avatar and his inferred hunger).
Kamikaze Midget, it is the extension to doing more than playing a role, and the end necessitating that means, that is key.
No, what is key is that a role-playing game involves playing a role.
Anything that is beyond that, anything that is MORE than that, isn't about whether or not someone is "role playing" anymore. Clearly, they are. Your concern is with whatever that "more" is, so define
that thing, and maybe we can all be on the same page.
Don't call it role-playing, 'cuz it really doesn't seem like it is.