They aren't functionally the same, though. In the former case, an author is creating a text (a semiotic object that conveys meaning), and more specifically a narrative text (i.e. a text set apart from non-narrative texts through the use of literary devices to connect a sequence of related events). Anything that can be read to discern meaning can be a text; and any kind of text can be marked as literary and therefore tell a story. Hence, a story can be a written tale, or an oral tale, or a silent movie with no words at all, or even a piece of music. ("For sale: baby shoes, never worn." —a very short story usually attributed to Earnest Hemmingway.)
Your latter case is only a story if it's an after-the-fact account of what happened during the game, in which case there's once again at least one author (whoever's telling the story) creating a text. The actual fact of gameplay is not the creation of a text, it's just a sequence of events happening in a fictional world. That's no different from a sequence of events happening in the real world, which aren't (and can't be) a story until someone chronicles them as history or journalism or biography or whatever.