Apparently I've already voted in this poll, which is either an ENWorld database hiccup or a Barsoomcore brain fart, because I have no idea what I might have chosen.
So given that I've already thrown off the statisical results, I thought I'd go through each item on the list. Because I can.
The honest and best answer is that I try to make sure everyone is having fun. The reason that's still the best answer, even given rounser's little query at the start of the thread, is because the method chosen to achieve that goal will vary from one game session to another. So rather than say, "I use this method most of all," which I couldn't say with any sort accuracy, I thought I'd say "I use this one and this one but not so much this one."
Verisimilitude, setting detail and staying true to setting
Sure, I use this method. To some degree. I like to have lots of details, but I also run pretty fast and loose, and handwave a lot of stuff that I don't feel like dealing with. And of course it varies tremendously from one game to the next. My pirate game had tons of setting detail. My stewardesses game, not so much. Both were hugely fun.
Presenting challenges for the intellect of the players
Yep. Puzzling my players is lots of fun. They've been known to create PowerPoint presentations to explain their theories on Barsimian cosmology. But then again, this varies not only from campaign to campaign, but from game session to game session. Depends on what I feel like doing, and what my players feeling like running with.
Campaign being most enjoyable at the level of each encounter
Campaign being most enjoyable at the level of each story arc
Campaign being most enjoyable at the level of each adventure
I don't really have much idea what any of these things mean, in reality. What's the difference between a campaign that's most enjoyable at the level of the story arc and a campaign that's most enjoyable at the level of each adventure? Isn't a campaign that's equally enjoyable at BOTH more enjoyable than one that's MOST enjoyable at one or the other? Maybe I'm missing something really obvious, but it seems to me that a campaign that's NOT enjoyable at the level of each encounter has a pretty low likelihood of being very enjoyable at the level of each adventure.
PC and NPC characterisation, relationships and roleplay
Critical part of fun stories, for me. Tightly tied into action and combat and problem-solving, for me. Of course, what constitutes characterization, relationships and roleplay varies from one game to the next. Sometimes bad guys just cackle and do naughty things, and sometimes they conduct elaborate schemes and pass themselves off as trusted allies for half a campaign before revealing their foul nature.
Non-linear gameplay and allowing for meaningful player choice
? I assume this means, "Not railroading." No, I don't bother strongly directing my players. My bad guys go about their schemes and if the players want to get involved, they're free to. Sometimes that results in not-very-linear stories and sometimes it results in high-speed express lines to Conclusionville. But in any event, I certainly would never tell a player, "No, you can't do that because I'm not prepared for that course of action." I would be far too humiliated.
Surprising players through plot twists and unexpected novelty
This is part of Presenting Challenges and PC and NPC Characterization. They try to figure things out, I shuffle things around. We all have fun.
Emphasis of a struggle versus evil and compelling villains
See above. Good bad guys require real characterization. They need to be smart enough to fool the PCs, stump them, confuse them and surprise them.
Presenting challenges based on the attainment of power
Not sure what this means. I think it's important to present challenges that are more-or-less matched to the player capability, because otherwise that's not lots of fun. But I'm not sure that's what's meant by "based on the attainment of power".
Emotionally involving players in the campaign
You're not having fun if you're not emotionally involved. "Emotionally involved" might mean different things in different games, of course. Is praying while you shake your die emotional involvement?
Kicking in doors, slaying monsters and taking treasure
Well, for a campaign that involves door-kicking, monster-slaying and treasure-taking this works great. Not so good for, say, a Call of Cthulhu game.
Presenting challenges based on drama and moral dilemma
Again, this is just so basic I hesitate to call it a priority. Good stories are stories that involve characters making tough choices. Whether those choices are "do I kick in the door or should I pick the lock" or "Should I tell the king about his son's treachery, or should I help the son because I think he'll be the better leader" is dependent on the type of game I'm running -- indeed, it varies from one session to the next in a single campaign.
So those are my responses. Hope they're helpful.