I don't think the second
definition helps us. I'll tell you why. Almost all
ethical theorists allow that the right or the good thing to do varies
with circumstances.
You gave the example of killing, and I think that's a good one.
Here's another: Aquinas held that stealing was wrong in normal
circumstances but permissible if you are starving.
There may be some moralists who have denied this sort of thing, but
not many. If this were all that ethical relativists were saying, it
would be old hat and there would be no controversy.
Ethical relativism cuts much deeper than this.