Immersive Story Game (ISG
) wouldn't cut it for me. An rpg without stats is pretty much a quest game, and when it came down to Kings Quest type games and Ultima (both of which I'd consider as having immersive stories), I was all over Ultima for hours and hours and days and weeks...
Thinking about the main differences in those two games in general: character building. From the ability to choose your gender and name, up to develop your overall characteristics like strength and intelligence, I had a lot more fun being a character who I could adventure with and have effected by those adventures.
The key definately wouldn't be to remove the stats existence, it would be to hide the quantified versions of them in exchange for qualified versions. Like in Grand Theft Auto, there may be a "fat" number, but its qualified as the character getting wider (there's also a bar quantifier, which would be better off gone for immersiveness).
Role Playing Games imply you play a role (by definition) so maybe a game where you decide the characters role and develop it to interact with the enviroment in a manner of your choosing should be called something different (it is more then simply playing a role, you are developping a character in an interactive immersive world [ideally]). Maybe CDG (Character Development Game) would better suit?
I was thinking about how a game with hidden stats would work (so that there are definately a complex set of statistics in the game, just not visible). You'd still want an involved character creation that is going to make your character excel in the areas you like (perhaps you want someone strong, or maybe intelligent, or maybe quick).
For anything physically reflected these could be decided by physical character creation, several sliders for muscle tone around the body (strong leg muscles for faster running, strong arm muscles for stronger weapon attacks or punches), and for fat stores (giving potential energy, allowing characters to easily build muscle from with work, or energy to expend on thought), bone sizes and proportions (perhaps longer arms for better reach, or longer legs for long strides), and then cosmetic changes which allow you to change your character to look more exactly how you want. Starting with a basic average being and having all those changes expend a bar that once fully expended allows no more changes (so on character creation you get to make some speed changes to your character to get it going in the direction you'd like and to personalize it), except through possibly adding to the bar by accepting some negative features (crippling effects, or maybe losing an eye "aarrgggh im a pirate" lol which allows you to make an even stronger character since you gave up a bit of charisma and depth perception).
Then would come the more abstract choices like intelligence. A series of questions like old style Ultimas or Elder Scrolls
aggerfall.
And then you could start the game and just enjoy yourself and the world.
Once in the world the skills you use a lot would develop and those you don't use would dwindle (and your physical appearance would respond accordingly), so that you could just play your style and become better in that style, and get to interacting with the world and the story.