I don't think that you properly described the Trinity.
Historically, when we in the Christian church have spoken of the Trinity, we are speaking of the one God. This is important, to understand that the original and major emphasis of the Trinity is to affirm the unity of the one God. That despite the experience of being taught by Jesus to pray to the Father, and that one will be guided by the Holy Spirit, and that Jesus himself is to be understood as both Lord and God (see Thomas' comment when he first meets Jesus after the resurrection, John 20:28) that we don't have three different God's but still just the same one God.
So, to speak of God is already to speak of Jesus and the Holy Spirit. To call the Trinity God and anything is to be redundant. Muslims make the mistake of equating the concept of the Father with all of God, but we who are trying to speak of the Trinity ought not do so.
Also, though the Son has always been, and therefore God has always existed in a Trinity which included an eternal Father and an eternal Son, we cannot say that Jesus has always been. Jesus is the incarnation of the Son in a human body, but that incarnation does have a beginning point in time, which the Son does not.
So, the Trinity is not God, Jesus, and Holy Spirit. The Trinity is nothing more than and nothing less than a description of what we understand about God pure and simple. That God has eternally existed in community within himself. That community is known to us in the persons of the Father, Son, and Holy Spirit who are nonetheless just one divine being, i.e. God.
P.S. I know that many who have already posted here are quite capable theologians. So, I'm surprised that no one else caught the mislabeling of the Trinity earlier. Are you all sleeping?