Firstly, "tri-theism" is just a heterodox as modalism as it would indicate belief in three gods, rather than one God. Trinitarianism is strict monotheism.
Secondly, the problem of the word "person" goes back to antiquity. When Greek-speaking theologians used the word "hypostasis" the Western, Latin-speaking theologians rendered it as the Latin "persona" from which we Anglophones use the word "person". The use of the Latin "persona" was troublesome for exactly the reason you mention, which is why Latin theologians had to carefully define "persona" as corresponding to the Greek "hypostasis". "Hypostasis" doesn't mean "person" or "persona", not really, but rather addresses a fundamental reality of a thing, in this sense it speaks of (for example) the Father's Father-ness.
You will note that I have a tendency to use the word hypostasis and hypostases (the plural form) rather than "person" and "persons". That's because I think the word "person" carries entirely too much baggage that confuses rather than helps understanding Trinitarian thought. The use of "hypostasis" is taken directly from the ancient Church Fathers and is clearly defined theologically and does not carry presuppositional baggage.
-CryptoLutheran
It's not "strict monotheism" because you have THREE persons which "should" be seen as three gods. But it isn't to avoid polytheism. This is why so many find trinitarianism to be illogical and cannot understand it. Judaism and Islam has "strict monotheism". One God - NO divisions or other forms. Just one indivisible God.
I think it's a big silly to start giving multiple divine beings "Father-ness" or "God-ness". Either something is God or it is not. Judaism says, "Hear O Israel - the Lord our God is One." How that equals three, I do not know.
The fact that theologians have to play word gymnastics just to try and squeeze some form of meaning out of these terms which do not make logical sense and no layman could understand anyway just shows the implausibility of it all.
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