Well I'm relying on a pretty common-sense notion of "story", more-or-less what I learned in school. I've just looked at Wikipedia - the entry for "story" is a mere index, but it leads to this page:
Narrative - Wikipedia. The opening line is " narrative, story, or tale is any account of a series of related events or experiences" and then some way down the page there is a heading "Aesthetics approach" and under that heading is the following:
Narrative is a highly aesthetic art. Thoughtfully composed stories have a number of aesthetic elements. Such elements include the idea of narrative structure, with identifiable beginnings, middles, and ends, or the process of exposition-development-climax-denouement, with coherent plot lines; a strong focus on temporality including retention of the past, attention to present action, and future anticipation; a substantial focus on character and characterization, "arguably the most important single component of the novel" . . .; different voices interacting, "the sound of the human voice, or many voices, speaking in a variety of accents, rhythms, and registers" . . .; a narrator or narrator-like voice, which "addresses" and "interacts with" reading audiences . . .; communicates with a . . . rhetorical thrust, a dialectic process of interpretation, which is at times beneath the surface, forming a plotted narrative, and at other times much more visible, "arguing" for and against various positions; relies substantially on the use of literary tropes . . .; is often intertextual with other literatures; and commonly demonstrates an effort toward Bildungsroman, a description of identity development with an effort to evince becoming in character and community.
The technical accomplishment of Apocalypse World and Dungeon World (and some other RPGs that have comparable origins in terms of influences and schools of design) is to have set out methods of RPGing which will produce fiction that tends to exemplify the criteria of "thoughtfully composed stories" and hence satisfies the aesthetic demands of storytelling,
even though no one has to thoughtfully compose a story.
The particular points that I've emphasised are
narrative structure and related
temporality, and
character and
identity development. The different voices interacting, including a narrator-like voice, will tend to be produced by the mere fact of a conversation among multiple participants, one of whom has an asymmetric role in creating the fiction (ie the GM), although given the spontaneous and unedited nature of play this is perhaps less likely to be as strong a feature as the others I've mentioned, which the structure of play deliberately sets out to generate.
As far as "rhetorical thrust" is concerned, DW and AW are interesting in this respect. In DW, the use of "bonds" to establish various sorts of relationships between PCs, and then the GM's use of their moves to invite and sometimes even oblige the players to act in response to or in expression of those bonds, creates some sort of "argument for or against a position" which is different from the sort of neutrality that you (Lanefan) prefer in GMing. (AW uses different techniques from bonds in this respect.)
Intertextuality and the use of tropes are of course part-and-parcel of all RPGing, and I don't think DW or AW has anything particularly special to offer in relation to these.
A couple of thoughts about this.
First, if you don't want that sort of structure,
and yet you assert that pretty much any combat encounter exemplifies that structure, does that mean you don't want combat encounters in your D&D play?