I apologize in advance that this is a lengthy response. I also am not a Greek scholar nor am I very good with nouns, adjectives, and definitive articles. My response is based on my own very limited knowledge, a Greek Interlinear translation, and by what was posted by the OP. Take it for what it’s worth.
Part 1: If the Word is Jesus and was with God, it appears that Jesus is different from God because the verse says that the Logos was “with” God. How could Christ be that same God when it says that the Logos is “with” God? Who is this God who was “with” the Logos in the second clause? Obviously it is not Jesus Christ since they believe he is the Word. They could not be the same person because a person could not be “with” another person if he is the same person!
The Logos was “with” God (the Father) in the second clause, and the Logos is God (the Son) in the third clause. They are not the same person. They are two distinct persons in relation to each other, one is Father and the other is Son, yet they are one in substance, namely God. To demonstrate this Jesus says “I and the Father are one” (John 10:30). And the Kingdom Hall Interlinear expresses it even more clearly in the literal translation which reads: “I and the Father one thing we are.” Not one person, but two persons who are “one thing”, or as Catholics would say; they are one substance or nature.
In addition Colossians 2:9 reads: “For in Him (Christ), the fullness of deity (literally Godship), dwells bodily.” And again in Hebrews 1:3 “He (Jesus) reflects the glory of God and bears the very stamp of his nature…” or as the New World Translation puts it: “He is the reflection of his glory and the exact representation of his very being.” Again, two persons that bear fully the Godhead in glory and in their very nature or being (i.e. one substance, one true God).
After it has been established that the God being spoken of in John 1:1 is indeed the one true God (i.e. the Word was with the only one true God), it would be redundant to restate it again in the same sentence. The God being spoken of in both clauses is the one true God for there is no other. The difference is the persons being spoken of.
Part 3:2 Since the term God is a monadic noun (one-of-a-kind), it is always being introduced by the article in the Greek….Now when Greek does not use the definite article with a noun, that noun becomes much more like an adjective.
That may be true in many cases but definitely not all cases as can be seen from John 1:6 “there was a man sent from God” and from Romans 9:5 “God who is over all be blessed forever”. The definitive article does not precede “God” in the Greek, nor is “God” being used here as an adjective.
If the definitive article always refers to the One True God, than how does a non-Trinitarian explain Titus 2:13 which reads: “….the manifestation of the glory of
the great God and savior of us, Christ Jesus.” The non-Trinitarian Kingdom Hall New World Translation adds another definitive article so that it does not read that Jesus Christ is BOTH the great God AND savior. Their translation is as follows: “…manifestation of
the great God and **of the **Savior of us Jesus Christ.” Their footnote in the Kingdom Interlinear states this is added “to agree with the distinction between God and Jesus”. In other words, to force the translation to fit their theology that Jesus is not God.
While Jesus does indeed refer to the Father as the only true God (John 17:3), it must be understood that Jesus here is speaking in his humanity which is demonstrated by verse 5: “and now, Father, glorify me in your own presence with the glory which I had with you before the world was made” (John 17:5). In his humanity he will die, but he will retain the glory that he had from the beginning with the Father as God. This also demonstrates that Jesus has the same glory with the Father in heaven (see also Rev 5:13). In this context, John 17:3 cannot be used as a proof text against Trinitarian belief, rather it upholds it.