You are correct, TRex. Quin and quint should have the same classification. I wasn't aware of quint being used for quintuplets in North America - I just thought everybody called them quins.
So the question is, should they be common or rare? Where a common word is spelled differently in America and Britain, I have generally made both versions common - e.g. odor / odour. But where a word is mainly used in one region - e.g. British doss, American coed - I tend to make it rare.
I think quin and quint are more in the second category, and should both be rare.
As for Uniate and Uniat, they seem to be capitalised in all dictionaries and in the overwhelming bulk of usage.