I personally believe the name "Jesus" was spoken audibly in early Christian baptism.
This may be reflected in
James 2:7 in the Amplified Bible which says:
"Is it not they who slander and blaspheme that precious name by which you are distinguished and called [the name of Christ invoked in baptism]?"
Also by a comment by William Barclay in his comment on
James 2:7 in his notes:
"It is the rich who abuse the name by which the Christians are called. It may be the name Christian by which the heathen first called the followers of Christ at Antioch and which was given at first as a jest. It may be the name of Christ, which was pronounced over a Christian on the day of his baptism. The word James uses for called (epikaleisthai) is the word used for a wife taking her husband's name in marriage or for a child being called after his father. The Christian takes the name of Christ; he is called after Christ. It is as if he was married to Christ, or born and christened into the family of Christ."
page 67 of The Letters of James and Peter
However, some believe that in places like
Acts 2:38, 10:48 and others where the expression of being baptized in the name of Jesus Christ or in the name of the Lord is used, it means in the authority of Jesus or as authorized by Jesus. So, people who understand it that way go to see where Jesus authorized baptism and by whom and they use the words of
Matthew 28:19 as Jesus spoke them. To them that is being baptized in Jesus' name because they use the words He used with His disciples.
I've also seen
Acts 2:38 paraphrased as "be baptized in the name of Jesus who was anointed for the the forgiveness of your sins."
I have heard (so this is hearsay) of two UPC churches (one in Louisiana and one in Ohio) who use the following formula when baptizing a person, "In the name of the Lord Jesus Christ, I baptize you in the name of the Father, and of the Son, and of the Holy Ghost."