I asked precisely for this reason, but no examples or definitions were provided.
This is a similar problem as the one trying to classify playstyles. There are words out there but little agreement as to what they mean.
So, how about throwing out some concepts there and see if any stick.
Trad-Story. Here the game is played in a traditional D&D way but there is an expectation that the characters are protagonists. This is characterised by a reasonable chance to survive a fight/encounter. The campaign ends if a TPK occurs or they defeat the final boss.
The story is the interaction between the party and various NPCs and resolves as a series of arc where the party unravels the plots of antagonist NPCs that have come to their attention.
It is reinforced by game elements that increase character competence. Like powers, skills, more hit points and so forth, a good skill challenge mechanic.
It could be further enhanced by things like Fate Points: - to be cashed in, in the event of a TPK. that is, convert it to a capture or waking up in the battle site sans equipment or what ever.
Or by a re-roll currency like Inspiration.
Or by allowing the DM to make a hard move on the plot like capturing the party in an encounter where they were overmatched with out playing out the encounter. This would then yield the party plot coupons that could be cashed in to set up their escape.
An example would be the rogue cashed in her coupon to declare that she has lock picks hidden in the braids of her hair with out any pre-establishing fiction.
A more narrative/drama focused game could have mechanics relating to back story or other elements but I am not familiar with such games to suggest any category names or the mechanics to support them.
What I do not want in response to this post are blank statement that one does not like this but is the category name useful, do the suggested mechanics fit into this or should they belong to another category or another (an suggest a name for this).
Ideally these suggestion should promote story (for some value of story) and I leave it to those familiar with modern narrative type games to offer their suggestions and classifiers.