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Here's a contradiction I found:
If I'm wrong, then, on what bases do people decide wether to upvote or downvote a question?
Voting, really, is supposed to be about how useful for the site a particular post is. That does imply a bit of bias, of course, but consider that the first thing you mentioned, the fact that questions on most SE sites need to not be opinion-based, is about whether or not a question is on-topic for the site, not necessarily whether the question is a good question. A question can simultaneously be on-topic and not a good question (i.e. it isn't necessarily useful to the site in the long term). Comparing the need for questions to be fact-based and the inherit opinionated nature of voting isn't really an apt comparison as the first concerns whether a question is fit for SE, and the other has to do with its quality and usefulness for the future.
When we determine the topicality of a question, and thus whether it should be closed, we typically don't insert opinions into the mix. Whether a question is a duplicate, needing details or clarity, or is opinion-based, is typically either a yes or no. Determining whether a question is useful for the site is certainly up to a degree of opinion, and likely always will be. You'll just have to live with that. This is doubly true for Meta, as voting is used as a way to signal agreement and disagreement with feature requests.
When posting on SE, you should be trying to ensure that your questions are:
If you tick all of those boxes, you've certainly earned an upvote from me, personally.
Everything below is my opinion/experience, and what I often do. This shouldn't be treated as policy, but more how I think about voting
Tl;dr: people often decide based on their opinion of how good the question/answer is
Questions on most Stack Exchange sites have to be seeking facts and not opinions
Yep, that's correct.
Reputation in those same sites is based on people's votes, which are their opinions
Yes, but you are IMO missing something important. Votes (at least how I use them) is my opinion of how good the question/answer is. I don't think these two are mutually exclusive. Voting is always (somewhat) subjective, but it is (often) one's opinion how good a question (or answer) is.
I vote for a number of reasons, but the most common is on the tooltip (hover over the up/down vote buttons):
Upvote:
This question shows research effort; it is useful and clear
Downvote:
This question does not show any research effort; it is unclear or not useful
Of course, people also vote for other reasons (for one, sometimes people downvote a totally off-topic question). Sometimes people vote for other reasons too, of course. Sometimes people vote up a question purely because they find it interesting*.
*I'm not saying to upvote bad questions, nor to judge these reasons, but they are some reasons for voting up/down
