I suppose there's also a difference between American Puritan names and the ones - some of them laughable, some whole sentences - used by some politically conscious English parents in the Commonwealth period.
Trinity? I just think of
Trini Lopez, so might the Puritans also have associated it with
Roman Catholicism? As for
Unity, the only one I've ever heard of, and it's firmly fixed in my mind, is the Nazi-supporting Mitford sister and that's not a pleasing association.