Do you believe that the Son of God has always been a person?
Yes.
At the end of John 8, Jesus conveyed to the Jews that He and Abraham met in person. Was that the logos of the Father or of Jesus in the OT?
I presume that you’re referring to John 8:56-58. If so, I don’t agree with your premise.
“Abraham your father rejoiced to see my day, and he saw it and was glad.“
(Jesus, John 8:56)
Jesus didn’t say that he and Abraham met in person. He said that Abraham saw his day.
“All the previous lightening flashes pale into significance before the blaze of this passage. When Jesus said to the Jews that Abraham rejoiced to see his day, he was talking language that they could understand. The Jews had many beliefs about Abraham which would enable them to see what Jesus was implying. There were altogether five different ways in which they could interpret this passage.
(a) Abraham was living in Paradise and able to see what was happening on earth. Jesus used that idea in the Parable of Dives and Lazarus (Luke 16:22-31). That is the simplest way to interpret this saying.
(b) But that is not the correct interpretation. Jesus said Abraham rejoiced to see my day, the past tense. The Jews interpreted many passages of scripture in a way that explains this. ‘By you all the families of the earth shall bless themselves,’ and said that when the promise was made, Abraham knew that it meant that the Messiah of God was to come from his line and rejoiced at the magnificence of the promise.
(c) Some of the Rabbis held that in Genesis 15:8-21 Abraham was given a vision of the whole future of the nation of Israel and therefore had a vision beforehand of the time when the Messiah would come.
(d) Some of the Rabbis took Genesis 17:17, which tells how Abraham laughed when he heard that a son would be born to him, not as a laugh of unbelief, but as a laugh of sheer joy that from him the Messiah would come.
(e) Some of the Rabbis had a fanciful interpretation of Genesis 24:1. There the Revised Standard Version has it that Abraham was ‘well advanced in years. The margin of the Authorized Version tells us that the Hebrew literally means that Abraham had ‘gone into days.’ Some of the Rabbis held that to mean that in a vision given by God Abraham had entered into the days which lay ahead, and had seen the whole history of the people and the coming of the Messiah.
From all this we see clearly that the Jews did believe somehow Abraham, while he was still alive, had a vision of the history of Israel and the coming of the Messiah. So when Jesus said that Abraham had seen his day, he was making a deliberate claim that he was the Messiah. He was really saying: I am the Messiah Abraham saw in his vision...
To us these ideas are strange, to a Jew they were quite normal, for he believed that Abraham had already seen the day when Messiah would come.
The Jews, although they knew better, chose to take this literally...”
(William Barclay, The Gospel of John, Vol. 2, pp. 34, 35, 36)
So, Jesus is claiming to be the Messiah. As children of Abraham they should have believed him. They didn’t.
”The Jews said to him, ‘You are not yet fifty years old, and have you seen Abraham?’”
(John 8:57)
As I pointed out, Jesus didn’t say that he had seen Abraham. He said that Abraham had seen the Messianic age, long before the Messianic age actually arrived. The Jews, as they so often did, misunderstood him. In their anger and continued rejection of his messianic claim, they thought he was a madman claiming to be a contemporary of Abraham.
“Jesus said to them, ‘Truly, truly, I say to you, before Abraham was born, I am.’”
(John 8:58)
The unbelieving Jews thought Abraham was superior to Jesus. The opposite is true. Jesus was destined to be the Messiah reinforces his Messianic claims to be greater than Abraham; he was destined to be the Messiah promised by God before Abraham was born.
“Jesus has been emphasizing his Messianic claim. He does not say that before Abraham was born the logos existed, he says ’I am.’ It is Jesus the Messiah, Jesus the man whom the Father had consecrated to the Messianic work who speaks. Just before this he had spoken of ‘my day,’ which Abraham saw (John 8:56), by which we must understand the historical appearance of Jesus as Messiah. Abraham had seen this, virtually seen it in God’s promise of a seed (Gen 12:3; 15:4-5) and had greeted it from afar (Heb 11:13). And now it is this one who consciously realizes the distant vision of Abraham who says, ‘Before Abraham was born, I am.’ Jesus, therefore, seems to affirm that his historic personality existed before Abraham was born. If that be the case, then its existence before Abraham must be thought of as ideal.”
(G. H. Gilbert, The Revelation of Jesus: A Study of the Primary Sources of Christianity, 214, 215)