what does classifying a playstyle as situation first or backstory first help in regards to running any game better?
Seeing as those phrases are my coinage, I thought I might reply.
To begin with: many, many RPG texts are very unclear about how the game is actually to be played. The first RPG that I, personally, read that I could see was just coming out and telling me how it was to be played was Burning Wheel. Since then I've discovered older texts that do this too (eg Maelstrom Storytelling, Over the Edge). I've also come to realise that Gygax's AD&D and Moldvay Basic do this to a large extent, though they have some misleading trappings (about heroic fantasy etc) in their introductory material.
But when playing a system like AD&D OA (which clearly is not meant to be played in the Gygax/Moldvay style) or Rolemaster, In my experience you're basically on your own.
I read a lot of stuff - both official stuff for both systems, and other commentary - which told me that I should create a rich gameworld, lots of interacting factions, etc (ie a "living sandbox"), or strongly implied that I should to that by presenting it as a model. Relatedly, both OA and various RM books have lots of tables for determining (perhaps by rolling, perhaps by choice) arcs of events in the campaign world, clearly intended to provide both a backdrop to play and material for play.
I also read a lot of stuff that talked about how to design and adjudicate action in locations, which clearly took for granted eg that movement would be adjudicated by tracking distance moved on a map vs movement rates.
And then one gets to the actual moment of play and of adjudication, and the question arises,
what to do with all this prepped material? Eg, if one of my background events is
the assassination of the emperor, and my maps and movement rates tell me that
if, at this moment, I reveal a rumour to the players, then even if they travel to the capital at their fastest they can't arrive in time to stop the assassination as per my prepared timelines, what am I to do? Stick to all my prep, and have this dramatic event happen offstage? Or adjust my timeline?
At the same time, it's becoming clear in play that the players are into some stuff - eg the scheming of the Scarlet Brotherhood - but not other stuff - eg the border wars in Furyondy. Do I still focus on both in my prep and management of the sandbox? What if my random rolls for event generation reveal that all the exciting stuff is going to involve Furyondy and not the Scarlet Brotherhood. What if the players decide to try and infiltrate a Scarlet Brotherhood stronghold and I don't have it prepped - or haven't even thought about whether and where it might be?
My ability to deal with these questions in my own play was helped a great deal by having someone else - mostly Ron Edwards - explain what was giving rise to them, namely, a certain set of assumptions about how prepped backstory, framing of situations, and resolution of declared actions, would all fit together. Encountering systems like BW - with its Circles and Wises checks - made it clear, by
showing how rather than just
asserting that, that other approaches are possible. I was able to realise that the relationship between successful moments of play, and various techniques I'd used without necessarily noticing that I was using them, was not accidental but causal. And so I was able to become more systematic.
And now, when I GM, I have a set of conceptual and practical tools for thinking about how I use prepped backstory, and how I prep for situation, and how I adjudicate actions. For instance: I can now see clearly that if (i) I treat prepped backstory as fluid, non-binding suggestions and (ii) I adjudicate declared actions as automatic failures
on the basis of unrevealed backstory then (iii) I'm basically just fiatting failure or the possibility of success. No AD&D or RM book ever explained this to me. I also now have a clear way of thinking about
how to bring backstory into framing, so it becomes part of the established fictional positioning. I also understand much better than I use to how to incorporate player suggestions about backstory into framing and adjudication.
All these things have made my RPGing better.