Back in 1999, I thought that the Unifying Force meant the Force in it's entirety, while the Living Force was part of the whole.
But the introduction of the Cosmic Force confused the issue for me.
Science Fiction & Fantasy Stack Exchange is a question and answer site for science fiction and fantasy enthusiasts. It only takes a minute to sign up.Sign up to join this community
From the way I understand it (sources from Wikia articles on both), the main aspect of Unifying Force wasn't its duct tape aspect (as with the Cosmic Force) of the Universe, but that it unifies both the Light and Dark sides into a single Force.
Ironically, it seems that this is mostly a terminological cross-purposing/confusion:
Primary sources: Darth Plagueis, Jedi Academy Training Manual, Star Wars: The Clone Wars – "Voices"