The words "Narrative" and "Story" are often used interchangeably, but they are not synonymous.
A Narrative is a set of event-types and ideas, typically related and selected for thematic purposes but not necessarily sequential nor dramatic. It is open-ended. Narratives involve theme, word choice and emphasis, but exist without any need for Plot. A Narrative does not need a Structure; it just happens. Entire cultures can (and do) tell themselves Narratives. The Theory of Evolution is a Narrative. "Land of the Free, Home of the Brave" is a Narrative.
A Story is more directly tied to the concept of Plot and Character. It is a set of events with a Beginning, Middle and End, selected and arranged so as to convey a particular Meaning and/or Point of View. A Story, when it "works" (i.e. in a manner many people find acceptably resonant with one Narrative Structure or another) is the manifestation of a Narrative, in such a way as to support or refute it.
A Narrative functions as both a Template by which Stories are interpreted, and as a Filter determining which Stories will be accepted and which will be rejected or denied by adherents of that particular Narrative.
A Narrative Structure can be emergent, but if everything works right, what it emerges "into" is an already-existing (because archetypal) Narrative Structure, in either its positive (affirming) or negative (denying) form. A Story's Structure is called a "Narrative Structure" when we are speaking of the framework or formula rather than the content. That's because we're looking at a template. Joseph Campbell's famous monomyth is a Narrative Structure. So is Aristotle's dramatic arc. A Narrative Structure is the shape and purpose of a Story, absent the mundane, non-archetypal details.
A storyteller with a given Narrative Intention may pre-select certain types of events as being more probable, or pertinent for inclusion, than others, in the Story that unfolds. It is possible to mechanize and gamify this. It is also possible to "bake it right in" to the structure of the game itself.
So. Is there such a thing as a "Story-Driven" game?
A 100% "railroaded" game can be "Story-Driven". But other than that, no. Stories cannot actually drive events, because they are comprised of events. The Story is the events that are recounted when the important points are selected afterward for retelling, the Narrative is the set of Stories and beliefs it makes manifest, and the Narrative Structure is the arc, or "shape" of the dramatic curve that Story ended up taking. So if we're talking about typical RPG design at least, the words "Narrative-Oriented" or "Story-Oriented" would be more precise ways to express what's really going on.
Is there such a thing as a "Narrative-Driven" game?
Most trad RPGs are not, the way the rules are designed. Because ideally the Story is an emergent property of what happens at the table, and the Narrative cannot really be judged until it's complete. But because of the difference between Narrative and Story, it is possible for a game session to be Narrative-Driven or Narrative-Constrained, while the Story itself remains undetermined. The only requirement is that the Story elements as they unfold must either be guided by mechanisms adhering to the designer's intended Narrative Structure (e.g. the tilt in Fiasco, escalation in Dogs in the Vineyard), or bounded by them (seasons in The Quiet Year, continued survival in Dread).