You are thinking of nihilism (specifically error theory it seems - that there are no moral facts and people are wrong for thinking there are) because relativism (whether relative to subject or culture) doesnt deny that there are moral truths, just that they are only correct for the individual or culture that holds them.
Cultural relativism: The view that an act is morally right just because it is allowed by the guiding ideals of the society in which it is performed, and immoral just because it is forbidden by those ideals.
Ethical subjectivism: The view that an act is morally right just because (a) I approve of it, or (b) my commitments allow it. An action is wrong just because (a) I disapprove of it, or (b) my commitments forbid it.
Same book as the other comment of mine you replied to.
So, no, I didnt get it wrong. And the consequences I pointed out do follow from cultural relativism.