When a person is baptized (immersed in water) in Jesus name, the name of Jesus is called upon him.

If titles (Father, Son, and Holy Ghost) are called upon a person, they are not baptized in Jesus name.

We are to be baptized in the NAME of the Father, Son, and Holy Ghost, which is JESUS. (Matthew 28:19, John 5:43, John 14:26, John 17:6, John 17:11-12, John 17:26, Acts 2:38, Acts 8:12, Acts 8:16, Acts 10:48, Acts 19:5, 1Corinthians 1:13-15).

We take on His name Jesus by baptism. By baptism, we become called by His name. (Acts 22:16, Acts 9:14, Acts 9:21, Acts 15:14, Acts 15:17, Romans 10:13, 1Corinthians 1:2, 2Timothy 2:19, James 2:7, Galatians 3:27).

It is by the name of Jesus that we have remission of sins. (Acts 10:43-48, 1John 2:12, Acts 22:16, 1Corinthians 6:11, Luke 24:47, Acts 2:38). His blood is applied to us by baptism in Jesus name. (1Corinthians 1:13-15, Romans 6:3-4, Colossians 2:12-13, Revelation 1:5).

"Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost." - Acts 2:38

If a person is baptized in another way besides in Jesus name, we find in the Bible people were re-baptized in Jesus name.

"And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism. Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus. When they heard this, they were baptized in the name of the Lord Jesus." - Acts 19:3-5