It's been a long time since I've dealt with the subject of baptismal formulas and to be honest, it looks to me like this isn't even an issue anymore in my sphere of influence.
I haven't been in a church where this was ever a controversy. But I have had a previous time in my life, early on in my ministry, where I came across a lot of individuals among whom this was a controversy.
What are baptismal formulas?
Simply put, a baptismal formula what a person says when they baptize someone else. The problem is that certain denominations or churches say something different on the basis of their belief in the Trinity or their rejection of it.
Oneness Pentecostals, who deny the Trinity, are obviously not going to say, "I baptize you in the name of the Father and of the Son and of the Holy Spirit". Instead, they're more inclined to say, "I baptize you in the name of Jesus."
What Does the Bible Say?
The fact is that Matthew 28:19 gives the command to be baptized in the Trinitarian formula. Those who hold to "Jesus name only" baptisms would argue that Matthew 28:19 isn't Trinitarian or something else.
They then would argue that Acts 2:38 shows Peter preaching to people to be baptized in the name of Jesus and doesn't mention the other members of the Trinity.
So Which One Is It?
There's been an obvious solution to the fact that one text seems to use a Trinitarian formula and another one calls for a Jesus name only. And it's one that I've been surprised to find out isn't as popular.
Most commentators rightly point out that Peter isn't appealing to a baptismal formula but is using the terminology of, "in Jesus name" as an appeal to the authority of Jesus.
Think about a messenger who comes, "in the name of the king". The idea is that the messenger carries the authority of the king.
And while this is true, there still is one more feature of the two passages that is even more glaring.
In Matthew 28:19, Jesus is telling a group of disciple makers to go out and make disciples. And the way they do that is by baptizing them in the Trinitarian formula. What Peter is doing is appealing to the authority Christ gave the disciples to make disciples. But the difference is that Peter is telling recent converts the reason why they should be discipled. In other words, a disciple maker is now telling future disciples why they should be baptized.
To summarize, Jesus is telling disciplers how to make a disciple. Peter, a discipler, is telling disciples why they should be baptized. And thus, both texts aren't in conflict with each other. And a baptizer should baptize using the Trinitarian formula. And the authoritative reason why a person should be baptized is that Jesus commanded it.