I take relativism to be the view that the truth-value of a token claim varies depending on who assesses it. This should be distinguished from contextualism, according to which different tokens of a sentence type may have different contents, say depending on the speaker. Indexical content, for instance, is context-dependent in this way. But the resulting claims are typically not relative, because any token utterance will be absolutely true or false, no matter who assesses it. “I am RC” may be true when spoken by me but not you, yet even you must agree that my token utterance is a true one. So it is not 'relative' in the above sense. Similarly, whether bright colours (for example) enhance an artwork may depend on context, i.e. the rest of the work. But this aesthetic quality will only count as a relative value if a token instance of it varies in value across viewers – say, if two critics can, without error, disagree over the qualities of a single painting.
Similarly for ethics: any adequate theory must be sensitive to the morally relevant features of a situation. It would be morally obtuse to claim that lying (for example) is always wrong, no matter the specific context. But this isn't relativism, so long as we agree that there's an objective fact of the matter in any particular case. Some lies are permissible and others aren't; but there's no one particular (token) act of lying that is at once both right and wrong, "relative" to different observers.
Note that, on this understanding, the claim that people ought to abide by the norms of their culture is not actually a form of relativism. It's presumably an objective fact what your cultural norms are, after all. So even if someone in a different culture would - due to the change in context - be bound by different norms, they won't (if they accept the above claim) dispute which norms apply in your particular situation.
Cultural relativism should be understood differently. In contrast to the above "cultural command theory", relativists will claim that cultures have no special authority over their members. They simply provide one standard of assessment, and others may provide alternatives, and there's nothing to decide between them (after all, any such assessment would itself be made from some arbitrary perspective or other). The mark of relativism, recall, is that one and the same particular act merits conflicting assessments. It is both right (from one viewpoint) and wrong (from another).
[Of course, we should reject both these views. Arbitrary opinions don't magically become true simply because they're endorsed by "the culture" in general. And, contra the relativist, some perspectives are more reasonable than others -- differences aren't necessarily arbitrary.]
In sum, I agree with Peter that "there are some objective facts about societies that ethics must take into account when making recommendations as to how individuals should act." Context matters, and cultural context is part of that. But I don't think we should consider this any kind of relativism, "bounded" or otherwise.
[See also my old post on 'objective moral relativism'.]
Update: I should clarify a respect in which the quoted analogy to contextualism may be misleading. The semantic content of an indexical like 'I' depends on who utters it. The respect in which morality is context-dependent is not semantic in this way. The semantic content of 'morally right' doesn't vary. It is rather a substantive fact that the term extends to some tokens of a type but not others. See my more recent post: Moral Principles, Objective Generalizations.