I think the problem lies in your usage of the phrase “objective fact”.
For example, if I claim “broccoli is tasty”, my claim purports to report a fact. Plausibly, it purports to report a fact about me—namely, that I like broccoli. If someone else were to claim “broccoli is tasty”, her utterance would also purport to report a fact—plausibly, the fact that she likes broccoli. So two token utterances of the very same type may pick out different facts. If this is the case, “broccoli is tasty” is true when asserted by broccoli-lovers and false when asserted by broccoli-haters. This should not be surprising, provided that it is interpreted as a disguised indexical claim.
Clearly, there is no experimental process whereby all right-thinking people can conclude that broccoli is tasty (or, alternatively, that broccoli is not tasty), even though several right-thinking people can justifiably arrive at this conclusion (by eating broccoli and liking it, say). Crucially, this conclusion is consistent with being a realist about broccoli-tastiness, but inconsistent with thinking there are objective facts about broccoli-tastiness (as you use the term). Likewise, one can be a realist about morality without thinking there are objective facts about morality (again, as you use the term).
Of course, one is free to interpret “moral realism” as you do—it’s a natural enough interpretation, and may even be the most common one among philosophers. However, this is not the definition given in the SEP. According to it, “moral realists are those who think that...moral claims do purport to report facts and are true if they get the facts right”. This does not entail that moral realists think that moral claims purport to report objective facts. But isn’t such a loose interpretation of “moral realism” vacuous? As you say:
The moral anti-realist can choose from among two main alternatives if she wishes to deny moral realism, which I understand as being committed to the following two theses: (1) moral claims purport to report some (not necessarily objective) facts, and (2) some moral claims are true. 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. 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!”