This is an interesting point because there's a lot of assumptions and expectations that can rear their head. As a DM, I run sandbox-y, where there's always more hooks than the players can follow, and they can (and do) go off an do something completely else. 100% homebrew.
Another DM in the same group also runs homebrew, and usually has a "here's your hook and adventure", but is game if we go off course to improv - though he may ask for a few minutes of prep time before doing something like a big improved fight to make it interesting, which is completely fair and for our benefit.
A third DM does an excellent job running adventures, and is up for lots of improving within the adventure, but expects that we'll be following the adventure put in front of us. Maybe not as the writers expected, but staying on goal.
And all three of our DMing styles come out in play. A good example was in a campaign run by the second DM where to where we wanted to go, we needed to promise something powerful to a faction we didn't support. There had been a number of moral choices in this campaign and one character drew the line at this. From my perspective that's not only fine, but the point of putting moral choices in. From the third DM, that was the next set to move forward in the adventure so we needed to take it. We were both right - if we were running our own campaigns. But what was right for this campaign depending on this DM. And that was a good lesson to be reminded of.
(BTW, the third DM does put morale quandries in his games. They are usually for bonus power, more loot, or something like that, so turning them down doesn't derail the plot. He's a hell of a lot of fun to run with and to run for even if we have different styles in this one aspect.)