At first, when the name of the Father and of the Son and of the Holy Spirit was invoked in baptism or in other contexts it was not affirming a belief in later Trinitarian dogma but was an affirmation of the old roman symbol (a primitive version of the apostle's creed). It is not making a statement that these are three persons of one God, but that these three are critical aspects of our confession of faith.
Early Christians, who were baptized in the name of the Father and of the Son and of the Holy Spirit were just making an affirmation of the Apostles Creed (not a belief in the doctrine of the Trinity): As you can see we have Father, Son, and Holy Spirit in the creed—but no conflation of the three. Jesus is the name of the Son, by which we are reconciled to the Father and receive the Holy Spirit. According to Acts 2:38, if we repent (towards God the Father), and are baptized in the name of Jesus (the Son), we will have the forgiveness of our sins and receive the gift of the Holy Spirit.
Acts 2:38 - Revised Standard Version (RSV)
<38> And Peter said to them, "Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit.
This is the meaning harmonized with Matthew 28:19.
For more on why baptism is in the name of Jesus, see https://baptisminjesusname.com
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