This also shows the harmony between Matthew. 28:19 (in the name of the Father, Son, and Holy Spirit) and Acts 2:38, etc. (in the name of Jesus).
The will or authority of the three individuals is exactly the same, for they are completely in harmony and united in will. What one authorizes is what the others authorize. What one says to do is what the others say to do. To act by the authority (in the name of) one, then, is to act by the authority (in the name of) all three, for it is the same authority. It is not that there is just one individual (Jesus) who wears three different titles, but there are three separate and distinct individuals in the Godhead all of whom authorize the same act in the same way.
An ambassador might say he acts "in the name of the President of the U.S.A.," or he might say he acts "in the name of the President, the Congress, and the people of the U.S.A." The two statements are both correct, but they do not say the President, Congress, and people are all the same individual. They say that he acts by the authority of all three. Yet the will of them all is viewed as one and the same will.
So "in Jesus' name" is not a formula for what must be specifically stated when baptizing a person. "In Jesus' name" simply means by Jesus' authority or power. To baptize in Jesus' name simply means to do so in obedience to His power or authority. His authority is the authority of God or Deity, which is the same power as that of the Father and the Holy Spirit. So to baptize in accordance with Jesus' name is to baptize according to His power or authority, which is the same as baptizing according to the name or authority or power of the Father, Son and Holy Spirit. See Matt. 28:18,19.
For in him dwells all the fullness of the Deity bodily Colossians 2:9