The phrase 'Story gaming' is as screwed up as the phrase 'trad gaming', because different people mean different things by the same names. For some people, story means something akin to literature, with a certain structure and thematic meaning. I have found that by story other people mean the immediacy of the moment, which is totally opposite. For now, let's call the first 'story gaming' and the second 'experiential gaming', as the experience of roleplaying itself is the goal, not the resultant story. I'll get back to experiential gaming some other day.
Now "story gaming" is different from "story games", just as "trad gaming" differs from "trad games". You can game for story perfectly well with trad games, and people have been doing that since RPGs were invented. It requires a different attitude, generally cooperative from the group - everyone needs to be on the same page. It is done on the Group level, not the Designer level. Trad gaming just encompasses both story and experiential gaming, as well as other playstyles.
I tend, like Levi, to define things from the center rather than by the edge. For me, the center for story games revolves around, but is not in any way limited to, the following: Shifting from character to player level frequently during the game, Conflict resolution rather than task resolution, Setting stakes before resolution rather than using random quality of success after resolution, and generally a view of the character as a pawn - the controls are at the Player Level.
Trad games tends to center for me around, but not at all limited to, these concepts: Staying at character level during play and at player level between play stretches, Task resolution as opposed to Conflict resolution, Quality of success determined after or simultaneously with chance resolution, and a general view of the character as avatar - the controls are at the Character level.
Games vary in how they are presented, how they are designed, and how they are played - both individually and as a group. A game might be presented as one thing, but be designed to do something else altogether, and may be played differently to either - both WoD and CoC are famous for this. Designing tends to set a bias - playing a game is easier if you run with the bias rather than against it. The bias reflects the designer's intent for the game.
The designer creates rules to channel play. This is Designer level stuff - rules, setting background, and the like. The designer can always elect to push aspects of the game to the Group level, either by making tools for group use or by not specifying/designing a rule - when stuff comes up in play not covered by rules, the group must thus make a ruling and go with it. The group can also always usurp control from the designer by over-ruling the rules. This body of rulings, both pushed by the designer and usurped by the group, is called 'House Rules', and will always differ from group to group.