I sometimes make a point of distinguishing story from narrative. I got the idea form a English Ph.D. candidate friend of mine: story implies a certain kind of structure, with a beginning, middle, and end, and that story has a point beyond being a bunch of words. A storytelling game is trying to create a story with the aid of game rules - but the goal is to end up with a story.
A narrative is just a bunch of stuff that happens presented in some form. I.e. a conversation about what happens to characters. A non-story ttrpg still has a narrative, but the narrative can be built without regard to story structure, themes, or any of the things that make a story more than a bunch of stuff that happened. An actual story may arise from the chaos, but it does not need to be the goal. (which is clear when you make the distinction.)