How is what you're describing different from
Tritheism (which is not Trinitarianism)?
It's grammar. Tri-nity, un-nity.
Theism has to deal with god. Trinity just describes the combination (or state of) of three things: father, son, holy spirit in this example. In itself, grammatically, it's not an religious word.
Like virgin vs virginity.
As nouns the difference between virginity and virgin is that virginity is the state of being a virgin while virgin is a person who has never had sexual intercourse, or sometimes, one who has never engaged in any sexual activity at all..
So you can say to protestants, father, son, holy spirit is a unity. On catholicism (which I think the word cones from), it's not a unity. Mass shows their relationship rather their nature as one another.
It's speaking of the nature of three things.
Another example
Trinity sentence examples. The Trinity--the three elements of matter--are sulphur, mercury, and salt. It seems clear that the trinity of Anu, Bel, and Ea in the old Babylonian religion has its counterpart in the Mandaean Pira, Ayar, and Mana rabba.
The three parts, each concentrating on a single aspect of his interpretative trinity, pose major organisational problems : repetition is hard to avoid, as is a measure of apparent internal contradiction.
Some people are against god, son, spirit bring three as one. Their nature has no comparison. It's not a unity, they translate. Pro-trinitatians use tri/unity but I think unity is a better word since there is only one god they believe not three.
semantics