Subjective Morality?

[The Grand Nudger]

That's the rub, no..there isn't.  The very minute that a person refers to our opinions or ideas they have firmly come down on the side of cognitivism.  If they think that at least some of our moral propositions refer to opinions that we hold, then they are describing -themselves- as cognitivists...and because of this, any non cognitivist objections are rendered inconsistent and made either in ignorance or bad faith.
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OK.  So, I'm beginning to understand that you see this as 'sides'.  And that 'moral facts' are dogma that one can only accept once one has subconsciously chosen a side (deduced from post #263).  Is there a group name for these sides (e.g. sect, cult, religion, tribe, team, position, school, whatever)?

They're called meta ethical theories or systems.  Trying to get at the ontology of morality.  As opposed to applied or normative ethical theories, or moral epistemology.  

And what is the tribal name (team name / side etc.) for one who holds that morality is both cognitive and non-cognitive?

Irrational and/or confused would be that tribes name.  The positions describe baseward exclusions.  If you think that our moral propositions express a state of belief, you have accepted cognitivism and rejected non cognitivism as a position, but you can obviously still accept that some moral propositions are properly non-cognitive.  A noncognitivist cannot make this distinction or accept this as a fact and remain a non-cognitivist.  If our moral propositions do express states of belief (any of them) - then their position is roundly and completely false on it's own terms.

That person would be negotiating over how much of morality as practiced is properly non cognitive, not objecting to the fact that our moral propositions can or do express states of belief.