You're right, not all cognitivists are anti-realists. But some are, including Eliezer.
Indeed, realists are generally considered cognitivist. But my impression is that if a moral system is labeled cognitivist, the implication is that it is anti-realist. That's because realism is usually the top level of classifying moral systems, so if you're bothering to talk about cognitivism, it's because the system is anti-realist.
This is correct I think, but confusing. All realists are by definition cognitivists. Non-cognitivist is simply one variety of anti-realist: someone who thinks moral statements aren't the kinds of things that can have truth conditions at all. For example, someone who thinks they merely reflect the speakers emotional feelings about the matter (like loudly booing).
Of the anti-realists there are two kinds of cognitivists: Moral error theorists who think that moral statements are about mind-independent facts but that there are no such facts And moral subjectivists who think that moral statements are about mind-dependent facts. If what you say is true, Eliezer is one of those (more or less).
There seems to be a widespread impression that the metaethics sequence was not very successful as an explanation of Eliezer Yudkowsky's views. It even says so on the wiki. And frankly, I'm puzzled by this... hence the "apparently" in this post's title. When I read the metaethics sequence, it seemed to make perfect sense to me. I can think of a couple things that may have made me different from the average OB/LW reader in this regard: