The question appears to assume that "son of the living God" (a biblical phrase) is somehow a lesser term than "God the son" (a man-made phrase) however Trinitarians do not think that Peter's confession of Jesus as the son of the living God is an understatement of any kind.
Peter was not the only one to whom Jesus' true relationship within the Godhead was revealed. The critical difference is that Peter received the revelation, as evidenced by his confession, whereas the others, acknowledging and understanding the revelation, rejected it.
And this was why the Jews were persecuting Jesus, because he was doing these things on the Sabbath. But Jesus answered them, “My Father is working until now, and I am working.” This was why the Jews were seeking all the more to kill him, because not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal with God. - John 5:16-18
What those who were persecuting and trying to entrap Jesus clearly understood was that when Jesus called God his Father he was not identifying God's fatherhood in the same way as all those who came before him. Here are only a few examples of many harvested from this paper titled "The Fatherhood of God in the Old Testament":
You are the sons of the LORD your God. You shall not cut yourselves or make any baldness on your foreheads for the dead. - Deuteronomy 14:1
Do you thus repay the LORD, you foolish and senseless people? Is not he your father, who created you, who made you and established you? - Deuteronomy 32:6
For you are our Father, though Abraham does not know us, and Israel does not acknowledge us; you, O LORD, are our Father, our Redeemer from of old is your name. - Isaiah 63:16
God is portrayed as Father/creator who formed man, formed a people for His own, protected, disciplined, and blessed as a Father, and, as a Father, grieved when his sons and daughters reject him over and over again. This type of language and imagery is all throughout the Scriptures and Jesus would have raised no eyebrows at all employing language of this kind.
John however, under inspiration of the Holy Spirit, explicitly declares (5:17) that when Jesus says "My Father is working until now, and I am working" he is not using the term Father in this generalized, national way. Jesus is calling God his own Father; his "personal, individual* Father. The Greek literally says "The Father of me" and not "the Father", "your Father", or "our Father". It is clear in verse 18 that this is exactly what Jesus was understood to be saying.
Further clarity is given in that those who heard Jesus refer to God as the "Father of me" also understood that this phrasing placed Jesus in a different category than all others: God, the Father of me, is a claim by Jesus to equality with God, which claim his enemies very clearly understood for the Scripture says "This was why the Jews were seeking all the more to kill him" The highly debated phrase "only-begotten Son" is pregnant with this understanding.
In Mark 14 we see a similar scenario where Jesus claims God as his personal Father and murderous intent results:
And the high priest stood up in the midst and asked Jesus, “Have you no answer to make? What is it that these men testify against you?” But he remained silent and made no answer. Again the high priest asked him, “Are you the Christ, the Son of the Blessed?” And Jesus said, “I am, and you will see the Son of Man seated at the right hand of Power, and coming with the clouds of heaven.” And the high priest tore his garments and said, “What further witnesses do we need? You have heard his blasphemy. What is your decision?” And they all condemned him as deserving death. - Mark 14:60-64
"God the Son" is a man-made theological construct intended to capture, preserve, and focus upon what Jesus was claiming and what folks were hearing and either receiving or rejecting: "... he was even calling God his own Father, making himself equal with God". When Peter confessed "You are the Christ, the son of the living God" this is what he was confessing; that Jesus is not a created son like every other person but he is God's begotten Son and equally God, just as Jesus claimed.
