Acts says Peter and Phillip baptize in the name of Jesus. Is this just shorthand, or was the early Church still wrestling with the idea of the Trinity ??
Baptism in the name of Jesus is mentioned a few times in the New Testament;
Acts 2:38," 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 will receive the gift of the holy Spirit."
Acts 8:16, “they had only been baptized in the name of the Lord Jesus.*”
Acts 10:48, "He ordered them to be baptized in the name of Jesus Christ. "
Acts 19:5, “When they heard this, they were baptized in the name of the Lord Jesus.”
The wording mentioning baptism in the name of Jesus only occurs only in Acts. And even there it is not consistent. Are we to be baptized in the name of the Lord Jesus or in the name of Jesus Christ? In each case it is used as a description of what was received, it is never put in the mouth of someone as a baptismal formula.
So why was this description used? In the 1st century AD there were several forms of baptism. In Judaism baptism was used by various prophets and preachers for various symbolic purposes. Even John the Baptist had a form of baptism (Acts 1:5; 1:22; 10:37; 11:16, etc). The Apostles were making a distinction that Christian baptism was different from these other forms of baptism. Baptism that was given to the followers of Jesus was unique and special.
The only wording in Scripture remotely constituting an actual formula for baptism comes from Jesus:
Matthew 28:19, “Go, therefore, and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the holy Spirit…”
This understanding of baptism as Trinitarian is practically unanimous in the early Christian texts. You can read more about that here:
CAF Tract - Trinitarian Baptism