I suppose I gravitate towards fun more, but that doesn't mean I won't end up the best anyway. That's just how awesome I am, hurr hurr hurrr. Sometimes though, trying to be the best at having fun is a thing. So I try to mix it up. Take for example a game I play all the damn time; Team Fortress 2. If you want to be the best, you choose the best class and the best weapons and you do your best to be the best. You see these people all the time. They're maining soldier or demoman or sinking hundreds of hours getting good at sniper, and everyone that's getting stomped by them is rolling their eyes thinking these tryhards are ruining the fun. I tend to agree. They're ruining it for themselves. I make a game of it to see how hard I can keep playing well while using worse classes or joke weapons.
Some people start hard, exert themselves as hard and as fast as they can, come, get bored, and go. I like to take my time. I like a little foreplay. Use some of the weirder toys, tease dominations, see how much fun can be had in as many different ways as possible, and always chase that new way to play. But some people want to play it like Ash Ketchum and just bag 'em all. I shake my head.
Ahem...
Can games have those 2 styles coexisting? Yes, many games do. And many games do suffer from developers intentionally dividing along those lines, or more often, trying to balance it both ways and falling short.
Many games try to do their best to please both aspects, but one of them usually ends up disappointing. Take for a very recent example Dark Souls 3. The invasion system is very heavily weighed in the favor of the invaded host intentionally to reduce the stress on the casual player, but this comes at great expense to those doing the invading, more so if what they assumed was that the standard invasion PvP would be more or less even as it was in the previous games.
Though that may not be a clear example. There are standard duels for battling on even ground, but the normal wilderness invasions exist in a state of limbo being non-consenting in a deliberate clash between the two styles of PvE and PvP. What's happened to that compromise mechanic is that it's become very weighted against the player that set out to be the best vs the one just trying to survive and keep their game fun.
"Compromise" has been made for the worse in this case.
You see the same thing in other games. MMOs have always had the standard "casual" player vs the "hardcore raider" so much so that the casuals very often never enter many of the raiding dungeons where the raiders play. In World of Warcraft they've tried their hardest to bridge that gap, but the unfortunate result they have is "tourist mode;" piss-easy raids that disgust some hardcore players. If this mode were a wretch I'd spit on them. Sure, many more players are having more fun and enjoying new content in their favorite game... but at what cost?!
Similarly there are those that like Structured PvP to be the best at, and I pity them so. They're playing in a game that has PvP as an afterthought.