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.