I will focus just on the two Persons/Beings, rather than the Three Persons/Beings (Eternal Heavenly Trio), since the OP is addressing just the Father and the Son.
1. The Father (only) is called "the only true God" in Jhn. 17:3. The words "the only true God" in koine Greek are, "τον μονον αληθινον θεον" (ton monon alethinon theon). The Father is never called "μονογενὴς" anywhere in scripture. If He had been, He could not be the Father of the Son.
2. The Son (only) is called "only-begotten", "God" (Jhn. 1:1c., "θεος ην ο λογος" (theos en o logos)) in (Jhn. 1:14,18, 3:16,18; Heb. 11:17; 1 Jhn. 4:9, is basically monogene (μονογενῆ; Jhn. 3:16; Heb. 11:17; 1 Jhn. 4:9) or monogenes (μονογενὴς; Jhn. 1:18) or monogenous (μονογενοῦς; Jhn. 1:14, 3:18)). The Son is never called, "τον μονον αληθινον θεον" anywhere in scripture. If He had been, He could not be the Son of the Father.
The word "true", from "αληθινον" (alethinon) means "original, underived."
The words "only-begotten", from "μονογενὴς" means "only one of the same nature" (ie. as from the Father's nature; nothing to do with time). The Sons' nature is of (from) the Father. The Father's nature is His own (original, underived). The Father's nature is never that of (from) the Son. For instance, my DNA (nature) is of (from) my Father. My Father's DNA (nature) is never of (from) me as his son.
The Son is not "the Father". Ever.
The Father is not "the Son". Ever.
The Father is "the Father" (Jhn. 10:36. &c).
The Son of "the Father" is also a "Father" (Heb. 2:13; Isa 8:16,18, 9:6; Jhn. 13:33, &c), made so by "the Father".
The words "Father" and "Son" are relational terms.
Who/what is "the word" of John 1:1? John says:
In the beginning refers to Genesis 1.
The Father, in Genesis 1 is the Person/Being identified as "God said ..." (speaking to the Son)
The Son, in Genesis 1 is the Person/Being identified as "God made ..." (obeying the will of the Father)
Light refers to Genesis 1, wherein the Son, is the Light that reveals His glory in Day 1, by request of the Father.
Word refers to Genesis 1, wherein the Son carries out all commands of the Father, as the Amen of the Father. Wherever the Son is, He says and does what the Father desires Him to say and do.
Thus
God (ie Person/Being of the Son) was with the God (ie Person/Being of the Father)
Deity was with the Deity.
Eternal Son was with the Eternal Father.
2nd Person/Being was with the 1st Person/Being.
Joh 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
Joh 1:2 The same was in the beginning with God.
Joh 1:3 All things were made by him; and without him was not any thing made that was made.
Joh 1:4 In him was life; and the life was the light of men.
Joh 1:5 And the light shineth in darkness; and the darkness comprehended it not.
Joh 1:6 There was a man sent from God, whose name was John.
Joh 1:7 The same came for a witness, to bear witness of the Light, that all men through him might believe.
Joh 1:8 He was not that Light, but was sent to bear witness of that Light.
Joh 1:9 That was the true Light, which lighteth every man that cometh into the world.
Joh 1:10 He was in the world, and the world was made by him, and the world knew him not.
Joh 1:11 He came unto his own, and his own received him not.
Joh 1:12 But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
Joh 1:13 Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God.
Joh 1:14 And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.
Joh 1:15 John bare witness of him, and cried, saying, This was he of whom I spake, He that cometh after me is preferred before me: for he was before me.
Joh 1:16 And of his fulness have all we received, and grace for grace.
Joh 1:17 For the law was given by Moses, but grace and truth came by Jesus Christ.
Joh 1:18 No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.