First, she can maintain that all moral claims are false, which is a plausible suggestion: perhaps our moral claims purport to be about some normative aspect of the world, but the world lacks this normative aspect.
That would still be discussing an objective claim - just one that happens to be false. On a part with discussing a mathematical proposition which is false, or an empirical hypothesis which is false: both of these are independent of the person who says them or believes in them. Just so, discussing normative aspects of the world - whether they exist or not, and whether they are as claimed or not - isn't the same as discussing normative beliefs of a person.
So calling this moral anti-realism seems to use my sense of "moral realism" (objective fact), not the SEP's.
Second, she can maintain that no moral claims purport to report facts; instead, all moral claims express emotions. On this view, saying "setting cats on fire is wrong" is tantamount to exclaiming "Boo!" or "Ew!"
In one way, this is again moral anti-realism in my sense of the phrase: the claim that morals don't exist separately from the moral beliefs of concrete persons. (I hold this view.)
In another way, it can be read as a claim about what people mean when they talk about morals. In that case, the claim is plainly wrong, because many people are moral realists.
So to sum up, I'm afraid I still don't see what it would mean to be a moral anti-realist in what you say is the SEP sense.
Continuing my quest to untangle people's confusions about Eliezer's metaethics... I've started to wonder if maybe some people have the intuition that the orthogonality thesis is at odds with moral realism.
I personally have a very hard time seeing why anyone would think that, perhaps in part because of my experience in philosophy of religion. Theistic apologists would love to be able to say, "moral realism, therefore a sufficiently intelligent being would also be good." It would help patch some obvious holes in their arguments and help them respond to things like Stephen Law's Evil God Challenge. But they mostly don't even try to argue that, for whatever reason.
You did see philosophers claiming things like that back in the bad old days before Kant, which raises the question of what's changed. I suspect the reason is fairly mundane, though: before Kant (roughly), it was not only dangerous to be an atheist, it was dangerous to question that the existence of God could be proven through reason (because it would get you suspected of being an atheist). It was even dangerous to advocated philosophical views that might possibly undermine the standard arguments for the existence of God. That guaranteed that philosophers could used whatever half-baked premises they wanted in constructing arguments for the existence of God, and have little fear of being contradicted.
Besides, even if you think an all-knowing would also necessarily be perfectly good, it still seems perfectly possible to have an otherwise all-knowing being with a horrible blind spot regarding morality.
On the other hand, in the comments of a post on the orthogonality thesis, Stuart Armstrong mentions that:
This is not super-enlightening, partly because Stuart is talking about people whose views he admits he doesn't understand... but on the other hand, maybe Stuart agrees that there is some kind of conflict there, since he seems to imply that he himself rejects moral realism.
I realize I'm struggling a bit to guess what people could be thinking here, but I suspect some people are thinking it, so... anyone?