"Objective morality" usually implies some outside force imposing morality, and the debate over metaethics (at least in the wider world of philosophy, if not on LW) is usually presented as a choice between that and relativism. If I'm understanding Eliezer's current position correctly, it's that morality is an objective fact about subjective minds. This quote sums it up quite well:

I am horrified by the thought of humanity evolving into beings who have no art, have no fun, and don't love one another. There is nothing in the universe that would likewise be horrified, but I am. Morality is subjectively objective: It feels like an unalterable objective fact that love is more important than maximizing inclusive fitness, and the one who feels this way is me. And since I know that goals, no matter how important, need minds to be goals in, I know that morality will never be anything other than subjectively objective.

Unfortunately, when people talk about "objective morality", they're usually talking about the commandments the Lord hath given unto us, or they're talking about coming up with a magical definition of "should" that automatically is the correct one for every being in the universe and doesn't depend on any facts about human minds, or they're talking about their great new fake utility function that correctly compresses all human values (at least all good human values, recursion be damned). I don't know how Eliezer feels about the terminology, but if it were up to me, I'd agree with advising against "claim[ing] an objective morality", if only so that people have to think about what parts of their arguments are more about words than reality.