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What the SEP actually says is, "Moral realists are those who think that, in these respects, things should be taken at face value—moral claims do purport to report facts and are true if they get the facts right," and that's it.
This is all a matter of misunderstanding the meaning of words, and nobody is objectively right or wrong about that, since the disagreement is widespread - I'm not the only one to complain.
To me, an unqualified "fact" is, by implication, a simple claim about the universe, not a fact about the person holding the belief in that fact. An unqualified "fact" should be true or false in itself, without requiring you to further specify you meant the instance-of-that-fact that applies to some particular person with particular moral beliefs.
If SEP's usage of "fact" is taken to mean "a fact about the person holding the moral belief", the fact being that the person does hold that belief, then I don't understand what it would mean to say that there aren't any moral facts (i.e. moral anti-realism). Would it mean to claim that people have no moral beliefs? That's obviously false.
On Eliezer's view, as I understand it, human!morality just is morality, simpliciter.
That's exactly what bothers me - that he (and other people agree with this) redefines the word "morality" to mean human!morality, and this confuses people (I'm not the only one) who expect that word to mean something else, depending on context. (For example, the meta-concept of morality, as opposed to a concrete set of moral beliefs such as Eliezer!morality or humanity!morality.)
I agree that if everyone agreed to Eliezer's usage, then discussing morality would be easier. But it's just a fact that many people use the word differently from him. And when faced with such inconsistency, I would prefer that people either always qualify their usage, or taboo the word entirely.
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When I say "objective fact", I mean (in context) a non-indexical one.
The original problem I raised was that some people who talked about things being "moral" meant those statements indexically, and others meant them objectively, and this created a lot of confusion.
I use the term "objective facts about morality" to mean "non-indexical facts which do not depend on picking out the person holding the moral beliefs". Moral realism is the belief such objective facts about morality can and/or do exist.
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!"