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.
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.
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.