That's a generic enough definition that it should get most RPGs. It also captures a lot of wargames, boardgames, and other things, though. Take Gloomhaven -- RPG or boardgame? Or Warhammer 40k, wargame or RPG with armies?
I'm not knocking the intent -- it's just a really hard thing to define RPGs. I've proposed definitions before that have fallen flat in the face of this or that example. I think that the best path forward is to offer a definition that works for the majority of RPGs but doesn't grab other games too much. IE, one that has grey areas of "shrug, maybe?" but doesn't get Monopoly involved. And, that's not easy, either.
What makes it tricky (or trickier) is that some games that really don't seem like RPGs (TO ME) are still at least toying with narrative/story telling. Gloomhaven comes to mind; my experience of it is a pure tactics game with lots of flavor, but apparently there are some people who think of it as an RPG, or at least there's discussion about whether it is one. FFG's various Lovecraftian cooperative games have gotten more and more story-ish, but they don't feel like RPGs (TO ME), and in fact the more story-ish they get the less I like them (because to the extent they feel like an RPG, they feel like an RPG with a bad GM). Anything that tries to get at narratives that emerge from play is going to leave out what has been termed a "choreographed novel" elsewhere, which might not be my preferred style of play but which I wouldn't say isn't an RPG.