I do believe you've nailed it incredibly well.
While I consider the UCC's formal corporate documents to confess a Trinitarian understanding of God, I would object passionately to actually using the term "Trinity" except in reference to the congregation where I was confirmed.
As my early pastors explained it, "The Trinity" is our best attempt to explain a mystery described in Scripture but not entirely within our ability to understand. Thus, while some of the outlines are clear to some, it remains a theological problem imperfectly solved. The same would be true of the Incarnation, Baptism, the Eucharist and such.
Inherently, it can not be a test of fellowship except for those who oppose it as totally without scriptural support.
If memory serves, none of the ancient creeds or confessions of the Reformation use the word "Trinity", either.