And Jesus came and said to them, "All authority in heaven and on earth has been given to me. 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, teaching them to observe all that I have commanded you; and lo, I am with you always, to the close of the age." (Mt 28:18-20)However, in the Book of Acts, we see that the apostles were baptizing people “in the name of Jesus Christ” (2:38; 10:48) and “in the name of the Lord Jesus” (8:16; 19:5). Are the apostles being disobedient? No.
Luke, the author of the Book of Acts, is not intending to be precise regarding the exact formula that was used, otherwise all four passages would agree. But they don’t. Sometimes the baptism is “in the name of Jesus Christ” and other times it is “in the name of the Lord Jesus.”
It makes more sense to say that Luke is simply differentiating the Christian baptism from the other baptisms of the day, such as John's baptism (cf. Acts 1:5,22; 10:37; 11:16; 13:24; 18:25; 19:4), Jewish ritual washings (Gk "baptizo"; cf. Mk 7:4; Lk 11:38), and the baptism for the dead (cf. 1 Cor 15:29).
We also can’t forget that, according to Matthew’s gospel, the command to baptize in the name of the Trinity was Jesus’ final word to His apostles before He ascended into heaven. They are a sort of last will and testament to them. This means that those words are very important, and you can bet the apostles hung on every last word of it. To think that they would then go forth and baptize in some other way is nonsensical.
Instead, because of everything that is at stake in Baptism, the apostles and their successors would have been very scrupulous regarding the manner in which it was to be done. And we see from the historical record that baptism was performed in the Trinitarian formula. For example, Origen writes in his Commentary on the Letter to the Romans (248 AD):
“The Lord himself told his disciples that they should baptize all peoples in the name of the Father and of the Son and of the Holy Spirit . . . for indeed, legitimate baptism is had only in the name of the Trinity"By baptizing as Jesus’ commanded, the Church is simply doing what Christians have always done.