I looked up the Wikipedia definition of Trinity you provided, and it seems to contradict itself. It says;
God exists as three persons or hypostases, but is one being, having a single divine nature
In this context, person and being are the same thing.(if person and being is different in this context, please explain the difference)
So to say God is 3 persons but one being is the same as saying God is 3 persons but 1 person.
The use of the word "person" isn't always helpful here, but it can be helpful in the sense that it refers to the concept of identity or "who-ness". It does have antecedent in the use of the Greek word
prosopon (plural
prosopa). The preferred language of the ancient fathers when talking about these things was to speak of the one
ousia and the three
hypostases.
The word ousia is generally translated as "being", "essence", or "substance", it stems from the Greek verb
eimi, "to be", hence its translation as "being" in English. It refers to a thing's
esse or "is-ness". In speaking about the Trinity this talks about what the Father, Son, and Holy Spirit
are, namely God. Trinitarianism goes on to say that the ousia is undivided and absolutely one. There isn't multiple instances of the divine ousia, there's only one instance of it, which is communicated freely between the Three. I want to touch on that after discussing the meaning of hypostasis.
The word hypostasis is a more difficult word, but can be understood as referring to the underlying reality of a thing; it is translated into Latin as
subsistentia and thus is sometimes translated as "subsistence" in English. Here the language is about the discrete and distinct "thises", there's the Father, there's the Son, and there's the Holy Spirit; each distinct as a "this".
We use the language of
prosopon or "person" because the Hypostases are personal and relational, there is act between them. And so, for example, the Father loves the Son, the Son loves the Father.
As I said, I would touch again on the communication of the divine ousia between the Three. Trinitarian theology begins by asserting the Divinity of the Father, the Father is God, the one God. It's why our Creeds begin by confessing, "We believe in one God, the Father Almighty". Theologians have poetically described the Father as the "Fount of Deity". So when we go on to speak of the Son and the Holy Spirit, we are speaking of these in relation to the Father, and when we say that the Son is God and that the Holy Spirit is God, it is because they share in the one indivisible essence of the Father. The Son is God because the Father is God; the Holy Spirit is God because the Father is God. So that the Father communicates His own essence, from all eternity, to the Son and the Spirit. This is why we speak of the eternal generation of the Son and the eternal procession of the Spirit. So that the Son has His essence from the Father as the eternally begotten of the Father, God of God; and likewise the Spirit has His essence from the Father as the eternally proceeding from the Father, and is therefore like the Father and the Son, truly God.
TL;DR version: Person and being are used to refer to entirely different concepts in Trinitarian theology; fundamentally the difference between "who" and "what".
-CryptoLutheran