What kind of evidence can you offer to demonstrate that you are right and I am wrong?
Who says I need "evidence" to argue that you should do something? I could rely on my perceived authority - in fact, you could take this as a definition of what "moral authority" is all about. Sometimes that moral authority comes from religion (or cosmology, more generally), sometimes it's derived from tradition, etc. So I have to dispute your claim that:
it's pretty much in the nature of a moral claim to be part of a debate or discussion.
since it is quite self-evident that many people and institutions have made moral claims in the past that were not perceived as propely being part of a "debate" or "discussion". It's true that, sometimes, moral claims are seen in such a way - especially when they're seen as originating from individual instinct and cognition, and thus leading people to think of themselves as being on the "right side" of an ethical dilemma or conflict. And yet, at some level, more formalized systems like law and politics presumably rely on widespread trust in the "system" itself as a moral authority, if only one with a very limited scope.
So, you're never going to get an answer to the question of "what a normative claim is", because the whole concept involves a kind of tension. There's an "authority to be followed" side, and an "internal moral cognition" side, and both can be right to some degree and even interact in a fruitful way.
I think there’s a confusion in our discussions of deontology and consequentialism. I’m writing this post to try to clear up that confusion. First let me say that this post is not about any territorial facts. The issue here is how we use the philosophical terms of art ‘consequentialism’ and ‘deontology’.
The confusion is often stated thusly: “deontological theories are full of injunctions like ‘do not kill’, but they generally provide no (or no interesting) explanations for these injunctions.” There is of course an equivalently confused, though much less common, complaint about consequentialism.
This is confused because the term ‘deontology’ in philosophical jargon picks out a normative ethical theory, while the question ‘how do we know that it is wrong to kill?’ is not a normative but a meta-ethical question. Similarly, consequentialism contains in itself no explanation for why pleasure or utility are morally good, or why consequences should matter to morality at all. Nor does consequentialism/deontology make any claims about how we know moral facts (if there are any). That is also a meta-ethical question.
Some consequentialists and deontologists are also moral realists. Some are not. Some believe in divine commands, some are hedonists. Consequentialists and deontologists in practice always also subscribe to some meta-ethical theory which purports to explain the value of consequences or the source of injunctions. But consequentialism and deontology as such do not. In order to avoid strawmaning either the consequentialist or the deontologist, it’s important to either discuss the comprehensive views of particular ethicists, or to carefully leave aside meta-ethical issues.
This Stanford Encyclopedia of Philosophy article provides a helpful overview of the issues in the consequentialist-deontologist debate, and is careful to distinguish between ethical and meta-ethical concerns.
SEP article on Deontology