I'm curious on the players who don't try to win. Are they simply indifferent about whether they win the game, or are they actively ignoring the victory conditions entirely?
For my, neither of the two alternatives you mentioned nails it. Actually, they sound to me like a play-to-win player honestly trying to understand what a don't-play-to-win player might think, but not really getting it yet since it's a concept too alien to him. No offense! You hopefully know how much I respect you and your approach, even though it's very different from my own.
Let me try to explain.
Personally, I've been a radical "play-to-win" player for more than a decade. In every game I started playing, I had to crank up the difficulty to the highest level, turn the game into the biggest challenge it could possibly pose, and then I wouldn't rest until I had beaten it.
However, about 15 years ago, this changed. I noticed that this "play-to-win" approach produced a gaming experience that wasn't actually fun for me. I like to play deep, complex games, and the AI of such games simply wasn't (and still isn't) up to the task. When I play these games "to win", the game tends to degrade into me trying to exploit the weaknesses of an AI that has been granted insane bonuses to give an illusion of a competitive adversary. This isn't fun at all. The whole atmosphere of the game falls to the wayside. Even a victory isn't really satisfying because I know I had to employ dirty tricks to overcome the insane bonuses of the AI.
After realizing this, I changed my playstyle, and I'm now enjoying games much, much more. In Civ games, the thing that I enjoy most is the creation of an alternative world history, with me being an active part of that. Of course I try to make my country strong, to keep it safe, to (sometimes, depending on which character I'm playing) turn it into the world's leading military, technological, or cultural force, etc. And the beauty of the Civ games (especially Civ4) is that winning the game coincides very well with such natural goals and decisions of the alternate-world leaders of which I'm playing one. However, when I'm faced with a decision to do something that I simply can't envision my character to do, just to improve my chances of winning the game, I simply don't do it.
For example: I often don't use slavery at all - I know it'd be a better choice strategically, but I can't see the leader character I'm playing advocating slavery. If I actually did use slavery in such a case, then I'd damage the "plausible alternative world history" atmosphere that I draw my enjoyment from. Another example: While I do enjoy winning the game, the moment when the victory is secured is rarely very special for me. However, I then tend to spend up to 8 hours stepping through the history replay. I massively enjoy seeing how the history of this world developed, and reveling in the details of developments that I couldn't even see while playing is something I'm usually immensely looking forward to.
So with regard to your options, I'm not indifferent to winning (as I said, I do like to win), nor am I ignoring the victory conditions entirely (I _am_ doing the things that will eventually lead to achieving one of them). However, I do these things not because I want to win the game, but because they are natural actions of the leader I'm playing in the given situation. I enjoy Civ games most when natural, historically plausible, in-character decisions are a good way to achieve victory. Civ4 was such a game in which this was very possible.