The Word is actually Logos in the original Greek, and applies only to the second Person (not the second part) of the Trinity. The concept of Person itself was defined with reference to the Trinity (see Gregory of Nyssa and the Concept of Divine Persons by Turcescu. No 'part' of the Father Incarnated: his only begotten Son did. The Father remained the eternal Father and the Son remained the eternal Son while both are Persons of the One Godhead along with the Holy Spirit.
The Son is the Logos of God Incarnate, as the Gospel of John so clearly states. Neither the Spirit nor the Father were Incarnate, although they remained Trinity and God. Only Jesus had Two Natures due to the Incarnation.
Maximus uses perichoresis to describe the interpenetration of essentially different natures. With the interpenetration, natures are utterly united but not altered qua natures. The wholeness of both the union and the distinction is strongly emphasized. Maximus is careful to point out that there is a peri-choresis but not a meta-choresis, that is, there is an interpenetration but not a change of one nature into the other.
Törönen, Melchisedec Unity and Distinction in the Thought of St Maximus the Confessor [Oxford Early Christian Studies, 2007, p. 122]
Working out the union and distinction of the Persons within the wholeness of God is the essence of Trinitarian theology and Christian faith.
kai theos en o logos
and God was the Word
We know that the Word is the subject because it has the definite article, as we translate it accordingly: and the Word was God. Two questions, both of theological import, should come to mind: (1) why was [FONT=Symbol, serif]theos[/FONT] thrown forward? And (2) why does it lack the article? In brief, its emphatic position stresses its essence or quality: What God was, the Word was is how one translation brings out this force. Its lack of a definite article keeps us from identifying the person of the Word (Jesus Christ) with the person of God (the Father). That is to say, the word order tells us that Jesus Christ has all the divine attributes that the Father has; lack of the article tells us that Jesus Christ is not the Father. Johns wording here is beautifully compact! It is, in fact, one of the most elegantly terse theological statements one could ever find. As Martin Luther said, the lack of an article is against Sabellianism; the word order is against Arianism.
Jesus Christ is God and has all the attributes that the Father has. But he is not the first person of the Trinity. All this is concisely confirmed in [FONT=Symbol, serif]kai theos en o logos[/FONT].
Wallace, Daniel B. quoted from Mounce, Willam D. Basics of Biblical Greek [Zondervan 1999, 2003 p27-28]