There is a lot of confused statements here, and what I was getting at does actually answer your question, albeit it's very brief and may not be clear.
To be baptised into the name of the Father, the Son and the Holy Spirit, and to be baptised into the name of Jesus Christ, is the same thing because there is only one God. There is one baptism for Christians, and this one baptism was instituted, not by man, but by God, or rather, the God-man Jesus Christ.
Jesus, by virtue of being fully man and fully God (not half and half, not a demi-god), is true God. He is the Son from all eternity, but also, incomprehensively, the whole Godhead dwells in Him bodily. So can we say that Jesus saves? Yes. Can we say that the Father saves? Yes. Can we say that the Holy Spirit saves? Yes. For it is one God, in three persons. God saves. "Jesus", quite literally means "God saves". So when Christ commands the apostles to baptise in God's name, they do exactly that - in the name of the Father, the Son, and the Holy Spirit - or, Jesus Christ. Furthermore, when we include "the Son", in the Trinitarian statement, we do name Jesus specifically as He is the Son. This is the one baptism that the Church has received.
Baptism is not our obedience to God, but rather it's the grace of God. It's not what we do for God, but it's what God does for us. It's His promise to us, that we will truly be baptised into His own name. This is apprehended through faith and effected through the power of the Holy Spirit.
Salvation is not a process. We are justified by grace through faith, once and for all, by the life, death and resurrection of Jesus Christ. Now, sanctification, on the other hand, is a process. The fruit of the Holy Spirit produces faith, trust, obedience and love for God, and joy, peace, patience, kindness, goodness, faithfulness, gentleness, self-control.