Just to muddy the waters further still, there is (or at least was) also Universalism as a Christian sect! Core message: Christ died to redeem all humankind (universal salvation), so stop worrying about saving your soul or anyone else's, and get on with following his actual teachings.

On another note, the International Council of Unitarians and Universalists, to which both the UUA and the CUC belong, admits explicitly-Christian member organisations. (As does the UK's national equivalent, going so far as to include "Christian" in its full name.)

Religion! It's generally a mess, but Unitarian-Universalism is extra muddy.