The idea that “in the name of Jesus” was something other than Luke’s short-hand for Father Son and Spirit was completely debunked by Derek Prince in his booklet “Burial by Baptism.” (1972)
It isn't short hand, as you say, but rather Biblical and the proper way to carry out the command by Jesus at the conclusion of Matthew 28.
In Matthew 28:19, Jesus does tell us to baptize in the name of the Father, and of the Son, and of the Holy Ghost. I'm not arguing that Jesus said that, because He certainly did. Those words are in red. However, he doesn't say "repeat after me" or nothing similar. Look at how the command is laid out; Jesus says "in the name of... and of" the Father, the Son, and the Holy Ghost, indicating that these three are One, and that only one name unites them.
That name is Jesus.
"I am come in my
Father's name, and ye receive me not." (John 5:43, KJV): Here Jesus says that he came in His Father's name, and since His name is Jesus, it is said here that Jesus is also the name of the Father.
"And she shall bring forth a son, and thou shalt call
his name Jesus: for he shall save his people from their sins." (Matthew 1:21, KJV): We know that the name of the Son is Jesus.
"But the Comforter, which is the Holy Ghost, whom the Father will send in
my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you." (John 14:26, KJV): Jesus tells us here that the Holy Ghost, which fell upon willing believers at Pentecost, would be sent in His name; that name being Jesus.
Jesus is the name the ties the three titles Father, Son, and Holy Ghost. At the conclusion of Matthew, Jesus lays out the commandment for water baptism by showing us the unity of the three titles in one liking: Jesus Christ.
Peter laid out the Plan of Salvation for the Church at Pentecost. He told us repent, be baptized in Jesus' name, and receive the Holy Ghost (Acts 2:38). (We know that every instance in which a believer was filled with the Holy Ghost, they spoke with other tongues as the Spirit gave utterance.)
Other examples throughout the Bible tell us that baptism is correctly performed in the name of the Lord Jesus Christ:
Acts 8:12 -- But when they believed Philip as he preached the things concerning the kingdom of God and the name of Jesus Christ, both men and women were baptized.
Acts 8:16 -- For as yet He had fallen upon none of them. They had only been baptized in the name of the Lord Jesus.
Acts 10:48 -- And he commanded them to be baptized in the name of the Lord.
Acts 19:5 -- When they heard this, they were baptized in the name of the Lord Jesus.
Acts 22:16 -- And now why are you waiting? Arise and be baptized, and wash away your sins, calling on the name of the Lord.
Baptism, as performed correctly by the early Church and many Christians today, is done by full immersion in water and in the name of the Lord Jesus Christ.