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.