I have often tried to tell people that “in the name of” in Acts 2:38, means or is "by the authority of." When I was looking at the Greek, it has the word “epi” in Acts 2:38 for the word “in” which means “on, upon”, and the usage is “on, to, against, on the basis of, at.”
In Matthew 28:19, it has the word “eis” for the word “in” and it means “to or into (indicating the point reached or entered, of place, time, purpose, result)” and the usage for that is “into, in, unto, to, upon, towards, for, among.”
Since Matthew 28:19 has “eis” for the word “in” and Acts 2:38 has the word “epi” for the word “in” does this change the meaning of Acts 2:38 when it says “in the name of”?
Each language uses prepositions in slightly different ways, so often a straight translation to one word doesn't always work. In this case, epi does mean "upon." Thus, in Acts 2:38, Peter is saying that people are to be baptized upon the authority of Jesus as Lord. Jesus' command becomes the foundation for why baptism is to take place.
In Matthew 28:19, Jesus used the word eis, which indicates a motion toward a particular point. Thus, in Matthew 28:19, Jesus is saying that baptism puts the person in alignment with the commands given by the Father, the Son, and the Holy Spirit. Or you could say that baptism is done for the purpose of fulfilling the command of the Godhead. Acts 8:16 and Acts 19:5 also use eis in the phrase "in the name of."
Acts 10:48 uses the Greek word "en" which refers to the place or state. Thus, Peter here is saying the command to be baptized is within the boundaries of authorized commands given by Jesus.
While there are subtle differences in the shades of meaning, in general, they mean basically the same thing.