We might define 'morality' in terms of the role it plays, or else in terms of its form or content. According to the former extreme (i.e. "neutralism"), a moral claim may have any content whatsoever. A person's moral principles are defined to be whatever that person takes to be overriding. The other extreme (i.e. "descriptivism") understands morality as having a distinctive form and content, e.g. it must be universalizable, impartial, and concerned with the promotion of human well-being.
What's interesting about this is how closely these conceptions track the internalism/externalism debate about moral motivation. Neutralism is clearly committed to internalism: moral beliefs are intrinsically motivating, because by definition our moral beliefs point to what we think we have most reason to do. Descriptivism, by contrast, seems committed to externalism. If a normative principle must have a particular content in order to qualify as a distinctively moral principle, then there's no guarantee that everyone will endorse such principles.
So, which option is better? I much prefer externalism. I can imagine someone sincerely saying, "Yes, I agree that X would be the right thing to do. But so what? I don't care about morality, I just want what's best for me, and I can better achieve that by doing Y." Upon hearing such a stark admission of amorality, I would probably be shocked and disgusted at the person's vicious character. But I would not consider their statement to be incoherent, nor would I think them necessarily irrational. So internalism strikes me as obviously false. Moral beliefs are not intrinsically motivating after all, one must also have the appropriate desires.
Note that neutralists must radically translate the egoist's statement above. They would interpret him as saying, "I agree that 'conventional morality' says X is the right thing to do. But that's just wrong. Really the right thing for me to do is look out for my own interests, and I can do that by doing Y. So Y is the right thing for me to do." This is certainly a possible statement to make, but it does not seem to capture what the egoist in the previous paragraph was saying. He genuinely thought that X was morally right - he just didn't care.
It is implausible to follow the neutralist in allowing moral claims to have any content whatsoever. This threatens to collapse all normative frameworks into 'morality'. We should not want to do this. We can distinguish between aesthetic, prudential, and moral values, based on their different contents. The egoist acts on prudential principles, not moral ones. If someone wants government to fund art galleries instead of helping the poor, this need not imply that they see art as having moral priority; rather, they might prioritize aesthetic value over the moral. These are distinctions worth making, but we can only make them if we define morality in terms of its content, and not just its practical role. Otherwise, non-moral value will be defined away, and its replacement so all-inclusive as to be vacuous.
There is a third option I haven't discussed here, and that is a content-based internalism. The idea here would be that moral beliefs have a certain content, in virtue of which all rational people would be motivated to act in accordance with them. This appears to contradict the belief-desire theory of motivation, but I think it doesn't need to. For consider the belief that "doing X will best fulfill my desires". It seems that such a belief would be intrinsically motivating to rational agents - precisely because it makes reference to one's desires. Morality is trickier though, because while we are [by definition] motivated to fulfill our own desires, the same cannot be said of other people's desires. There doesn't seem to be any necessary reason why we must seek to advance others' interests. So I don't find this sort of internalism any more convincing than the neutralist variety.