To the participants. In this case, that would be
@pemerton since we are talking about his preferences.
How about an example? Let's say that the PC is a fighter who is looking for his brother, who rumor has it joined some kind of cult and ran off. The fighter wanders the land trying to find a clue to his brother's whereabouts, so he can ultimately find and save him.
This is central to the character, right?
Is it central to play? That is the question.
Is the game about what the player wants the game to be about?
Someone like
@Lanefan might say absolutely not. He sees that kind of personal quest as being boring to everyone else at the table, and so it is self indulgent on the part of the player who'd like to see this play out. He specifically does not want this level of player agency in his game. He prefers that whatever agency is allowed is happening at the character level, with the player declaring the actions and decisions he'd like for his character.
To
@pemerton, he has specifically cued the GM to what he'd like to see come up in play. For the GM to ignore that and instead just run his prepped material, whether published or of his own design, would be frustrating. He wants play to be about his PC's search for his brother, and the related beliefs and principles that may be called into question by that search.
Does this mean that every single thing that happens in play needs to revolve around the missing brother? No, of course not. But for it to be meaningful (and I'd argue, objectively so), it has to matter more than the PC showing up in a new town, asking around about his brother, and being told "nope, never saw this kid around here" and then roleplaying sadness at the lack of news.
It has to matter to the unfolding fiction. A series of clues or sightings or rumors leading the PC on in his search, learning more and more until finally the situation boils to a head, and the brother is found, or the cult he joined is confronted, or what have you.
Now, I think a lot of the confusion about this simply comes down to the specific game in question, and what the expectations for that game would reasonably be. Some games are absolutely designed to deliver this experience. Others are not suited for it at all. I think in most cases, people will adjust their expectations according to the game they're playing.
I think where we find conflict is with games that are somewhat suited for it, but for which it is not a necessity. Most versions of D&D would fall into this category, I think. Can it be done in D&D? Sure, to an extent at least. I run a 5E game and it very much revolves around what the players want for their characters. Am I guided in any way by the game to do that? No, not at all, really. The PCs (sometimes) have Traits, Ideals, Bonds, and Flaws.....but they don't really do much, and the books don't really offer the DM much in the way of their use. It's more just about the player getting a slight perk for actually roleplaying their character.
Other games have similar character traits that are integral to play.