Yes, people who say that realists are cognitivists say that this is true by definition, but I don't think these terms are used consistently enough that it is a good idea to argue by definition.

I'm not trying to argue by definition: I'm just telling you what the terms means as they are used in the metaethical literature (where they're used plenty consistently). If someone wants to say they are a moral realist but not a cognitivist then I have no idea what they are because they're not using standard terminology. If someone doesn't fit into the boxes created by the traditional terminology then come up with different labels. But it's an incredibly confusing and bad idea to use an unorthodox definition to classify yourself as something you're not. You representation makes me more confused about Eliezer's views. Why position him with this language if you aren't taking definitions from an encyclopedia?

According to the standard groupings being an anti-realist cognitivist and objectivist would group someone with the error theorists. If Eliezer doesn't fit there then we can come up with a word to describe his position once it is precisely distinguished from the other positions.