I agree with the connection made by the OP. Without detracting from this, let add to it by providing further evidence.
John makes a similar connection at the start of his first epistle:
- 1 John 1:1, 2 - That which was from the beginning, which we have heard, which we have seen with our own eyes, which we have gazed upon and touched with our own hands—this is the Word of life. And this is the life that was revealed; we have seen it and testified to it, and we proclaim to you the eternal life that was with the Father and was revealed to us.
Of all the NT writers, John uses what in Greek logic are known as category statements more than anyone else and most of these have Jesus as their subject. That is, John teaches more about Jesus and God via category statements than any one else. The most famous category statement of John is found in 1 John 4:8, 16 where he famously asserts:
God is love.
Back to Jesus. The Gospel of John contains a series of category statements based on various well-known metaphors:
- “I am” the Bread of Life (John 6:35-51) - a reference to the manna of the desert wanderings as well as the shew-bread in the OT tabernacle.
- “I am” the Light of the world (John 8:12) - a reference to the menorah in the temple and the parade of lights, one of the Jewsish festivals
- “I am” the Door of the sheep (John 10:7-9) - see #4 below.
- “I am” the Good Shepherd (John 10: 11-14) – a reference to YHWH as the great shepherd in places like Ps 23 & Eze 34:11 etc.
- “I am” the Resurrection & Life (John 11:25) - this may allude to Job 19:25-27 but this is not clear.
- “I am” the Way, Truth & Life (John 14:6) - a constant theme of John. In John 1:18 he declares that Jesus is "full of grace and truth." John 17:17 also says that "Your word/Word is truth."
- “I am” the Vine (John 15:1-5) - a reference to a metaphor of the vine in the OT.
John 1:14 establishes that Jesus is the Logos/Word. John then has a series of assertions around this noted by the OP as more category statements:
The Word was God (John 1:1) and Your word/Word is Truth. (John 17:17)
But there is more. Jesus repeatedly claimed to be the "I Am" of the the OT referring directly to Ex 3:13-15 and its subsequent use in the LXX in places like Deut 32:39, Isa 41:4, 43:10, 13, 25, 45:19, 46:4, 48:12, 51:12, 52:6. (This is clearer in Greek that in English.)
- John 4:26 – “Then Jesus said, ‘I am.’” [To the Samaritan woman at the well.]
- John 6:20 – “But then [Jesus] said to them, ‘I am. Fear not.’” [To the frightened disciples in the boat.]
- John 8:24 – “If you do not trust/believe that I am, you will die in your sins.”
- John 8:28 – “When you will lift up the Son of Man, then you will trust/know that I am.”
- John 8:58 – “Truly, truly, I say to you; before Abraham existed, I am.” [The Jews then tried to stone Him for blasphemy.] Note that this and the previous two mean that Jesus, in the space of this chapter of John 8 uses the unpredicated “I am” idea in the present (v24), future (v28) and past sense (v58). V24 & 28 appears to be tied to believers’ salvation as well.
- John 13:19 – “From now [on] I tell you before the occurrence, that you may believe when it occurs that, I am.”
- John 18: 5, 6, 8 – “He said to them, ‘I am.’ …Therefore, when He told them, ‘I am’, they fell backward to the ground.” [This occurred when the Jews tried to arrest Jesus in the garden. It could be reasonably argued that this is a case of identification. However, the fact that the arresting mob fell backward suggests that much more is intended here.]
There can be little doubt that John intended to defend the divinity of Christ (as well as His humanity; but that is another subject) by both implicit associations and explicit declarations such as:
- John 1:1, 18, 5:17, 18, 23, 10:30, 20:28.
In the book of Revelation we have some further material: Jesus says: "I am the first and the last" (Rev 1:17, 18, 22:13) which is a direct reference to a title of YHWH/Jehovah in the OT, namely, Isa 41:4, 44:6, 48:12.
We also have John giving Jesus the title of "Lord of Lords" in Rev 17:14, 19:16 which is reference to another title of YHWH/Jehovah in the OT, viz: Deut 10:17, Ps 136:3, 26.