So in John 1, it says that the Word was God, and I am curious as to whether the Word of God is considered part of the Father, Son, or Holy Spirit? Or is it distinct, and if so, why is it not included in part of the Trinity?
Welcome, and thanks for your question.
The Apostle John made clear that the Word is indeed the Son. In the first chapter of his gospel, he frames the narrative to follow as the story of the Word of God becoming a man, Jesus Christ. Here are some verses from that text:
1 In the beginning was the Word, and the Word was with God, and the Word was God. 2 He was in the beginning with God. 3 All things came into being through Him, and apart from Him nothing came into being that has come into being. 4 In Him was life, and the life was the Light of men. ... 9 There was the true Light which, coming into the world, enlightens every man. ... 14 And the Word became flesh, and dwelt among us, and we saw His glory, glory as of the only begotten from the Father, full of grace and truth. ... 17 For the Law was given through Moses; grace and truth were realized through Jesus Christ.
So, when used in a trinitarian sense, the "Word" or "Logos" of God is synonymous with "the Son", each being interpreted to mean the person Jesus Christ.
John's gospel uses the Greek word Logos which is translated as Word. Logos is the Greek term translated as “word,” “speech,” “principle,” or “thought.” In Greek philosophy, it also referred to a universal, divine reason or the mind of God.
To make it absolutely clear who John is talking about, after saying (in verse 3) that through him all things were made, verse 14 goes on to explain:
The Word became flesh and made his dwelling among us.
Jesus, the Son, is the Word, or Logos, who was with God in the beginning and who condescended to lower himself and come to earth to be born of a woman. Jesus is God incarnate and part of the One Being of God.
What role does The Word have in The Trinity?
I would provide an answer from biblical standpoint although the Catholic Doctrine on Catechism explicitly expound the Mystery of the Most Holy Trinity but for simplicity the bible already have a very clear answer.
The "Will" of God the Father is communicated thru "Word/Logos" and understand thru the "Wisdom" of the Holy Spirit. The WILL, the WORD/LOGOS and the WISDOM is the symbol of the Most Holy Trinity.So, if you want to understand the Word you need the Wisdom of the Holy Spirit. The question is who represents the Wisdom in the flesh as the Word became flesh and the Will is accepted thru Mary's FIAT. The Word got the flesh thru Mary the Will was revealed thru Mary. Did Wisdom got also the flesh thru Mary or the Wisdom use the flesh of Mary to manifest the Will.
I posted the above comments and I will support it biblically.
1. The "Will" of God the Father is communicated thru "Word/Logos"
John 6:38 New Living Translation (NLT) "For I have come down from heaven to do the will of God who sent me, not to do my own will.
Jesus as fully man was given a rational soul with human will, but His mission is to do the Will of the Father.
2. and understand thru the "Wisdom" of the Holy Spirit.
John 3:8 New Living Translation (NLT) 'The wind blows wherever it wants. Just as you can hear the wind but can’t tell where it comes from or where it is going, so you can’t explain how people are born of the Spirit.”
So, to understand the Will of God communicated thru the Word of Jesus Christ we need the Wisdom of the Holy Spirit.
Wisdom 1:7 New American Bible (Revised Edition) (NABRE) "For the spirit of the Lord fills the world, is all-embracing, and knows whatever is said.
Wisdom of Solomon 1:7 (New Revised Edition) "Because the spirit of the Lord has filled the world, and that which holds all things together knows what is said,"
Therefore, a believer must acquired Wisdom first because without Wisdom one cannot understand the Will of God communicated thru Word of Jesus Christ.
Lastly, how about the Will of the Father? How can we know the Will of the Father, as knowing and following the Will of the Father is the requirement in gaining entrance to Heaven.
Matthew 7:21 King James Version (KJV) "Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.
In closing. the role of the Word of Jesus Christ is to fulfill the Will of the Father and we can only understand the mystery behind the Word if we possessed the Wisdom of the Holy Spirit.
What is the Will of the Father, it's not the question here, so let's better leave it like this...