One option would be to define 'moral reasons' as a subset of our normative reasons. For example, we might say that some reasons are 'prudential', in the sense that the reason exerts its force for the agent's own sake, whereas all other reasons will count as 'moral'. I guess that's okay as far as it goes, and maybe there are some contexts in which this distinction could be philosophically useful, but for the most part I don't really see the interest in such stunted normative concepts. Why philosophize about a merely "some things considered" ought? To restrict the content of the fundamental normative concept is also to restrict its interest.
A more intriguing possibility is suggested by Parfit: perhaps there is a second primitive (undefinable) normative concept, besides that of a normative reason, for which Parfit uses the phrase "mustn't-be-done". The thought seems to be that certain acts have the primitive feature that they mustn't-be-done, which may create - rather than merely signal - a (decisive?) normative reason against so acting. The question whether this generated reason is decisive is just the debate over whether morality is overriding -- a debate that seems less substantive on alternative conceptions of 'morality'.
Utilitarians (like Egoists) might be best understood as nihilists about 'morality' in this undefinable sense of mustn't-be-done. Rather than offering a 'moral' theory, they offer a normative theory to rival morality. As Parfit writes (On What Matters, chp 7):
These people may be convinced that it matters greatly how well things go, and they may be strongly motivated and often moved to act in ways that prevent or relieve suffering. But they may be doubtful whether any acts are duties, or mustn't-be-done, and doubtful about blameworthiness, and about reasons for remorse and indignation. That is one way in which this form of Consequentialism might be an external rival to morality.
Some questions: Can you make sense of the indefinable, substantive moral concept of "mustn't-be-done"? If so, do you think it is applicable, i.e. that some acts actually possess this feature? Is this the best way to distinguish 'moral' concepts?
I feel like I have a slippery grasp of the intended idea (it seems very deontological), but perhaps it should ultimately be abandoned as senseless, or at least inapplicable. It may be easily confused with the derivative notion of an act that is prohibited on indirect consequentialist grounds -- i.e. an act that we ought to rule out of consideration. But this isn't a new primitive concept. It instead derives from the ordinary normative concept employed by utilitarians, simply applied to decision procedures.