Another good question.
The point of your article, I believe, is that just because we don’t know the answer to a moral question, doesn’t mean there isn’t one. And just because we think we know the answer to a moral question doesn’t mean we aren’t caught in our own cultural relativism.
The first sentence is correct, but the second one is not quite right. In that second sentence, you’re using “cultural relativism” as a way of referring to a culture’s conception of morality. But cultural relativism is something else: it’s both a metaethical and a normative theory which says an act is right if, and only if, the culture in which it is uttered believes it is. So, while I and everyone else are influenced by our cultural traditions, that’s not the same as saying I and everyone else believe that actions are right when they are what our cultures believe. The latter is a normative theory; what you’re referencing in your second sentence is more like a fact from psychology.
But your broader point is a really interesting one. You want to know — if I’m understanding you correctly — how exactly we can know when something is right or wrong.
That’s precisely what a normative theory attempts to provide for you. Normative theories attempt to specify what we ought to do; they inform us about what morality requires, and as a consequence they help guide our actions. So when cultural relativism says that an act is right when a culture believes it, the action-guiding aspect is this: to do what is right, do that which is consistent with the culture’s ideals. Or try utilitarianism, another normative theory, which gives you this formula: an act is right if, and only if, it maximizes pleasure and minimizes pain.
What do we do with these formulas? We apply them, using them to guide our actions. How are these formulas justified? Philosophers give lots of reasons for their acceptance. For example, utilitarianism says we should maximize pleasure and minimize pain, and it says this because the theory considers these states — pleasure and pain — to be the most salient experiences when considering rightness and wrongness.
So, why is it wrong for me to murder someone but not wrong for a tiger to kill another animal? Because wrongness implies a capacity to appreciate moral considerations — something I possess but the tiger does not. And wrongness, for most thinkers, implies blameworthiness. But we don’t blame tigers.
Now, a related but different set of concepts from right and wrong is good and bad. Is it bad, from the deer’s standpoint to have been killed? Yes. But that’s not the same as saying the tiger has done something wrong.
This is one way we can evaluate the morality of thoughts. Many of them are involuntary, which means we shouldn’t be blamed for them. But we can still say they are bad thoughts to have.
What makes them bad will depend on your value theory. If you consider human life valuable and think it is worth protecting, then a murderous thought, while perhaps not blameworthy (in certain cases it would be blameworthy, such as when you entertain the thought at length or even fantasize about it), is nevertheless a bad one.