> By "agree with the content", the author clearly doesn't means "believe that the content is factually incorrect"
I disagree that that is obvious, and, moreover, I would disagree with the idea that the author would have any way of telling whether someone flagging merely disagreed with some aspect of the viewpoint/presentation outside of the base facts or disagreed with both the viewpoint and the fact claims, seeing the latter as misrepresented to serve the former. Further, it's virtually impossible to cleanly separate disagreement on viewpoint from disagreement over facts, because the two mutually interact (particularly viewpoints not only shape perception of facts but also perceptions of the; relative importance of facts.)
For instance, there is a story right now circulating in even fairly mainstream, especially conservative, media (notably including Fox News) and being widely shared on social media alleging that a proposed California law would ban restaurants from serving children any drinks except the two choices of milk and water. This story is invariably couches in a lot of viewpoint around crazy nanny state liberals and other conservative stereotypes of California. It is very likely that lots of people would disagree strongly with the viewpoint of the story.
But the actual proposal, as one discovers with a few minutes of research, (1) doesn't limit what restaurants can offer to children (it limits “default drink options” for children's meal combos) and the limitation it proposes isn't to just milk and water (it includes milk, water, flavored water, and nondairy milk alternatives). It is, in other words, factually false in several dimensions.
Now, if one agrees with the viewpoint, one may be more likely to dismiss those factual errors, even if you are aware of them, as minor and tangential. If one disagrees with the viewpoint, one is likely more inclined to see them as significant and central and indicative of a deliberate distortion to push an agenda.