He was baptized with John's baptism - yes. But that is not the same thing as "in whose name?". John's baptism was one thing, Jesus' baptism was another :
I indeed baptize you with water unto repentance. but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire:
Matthew 3:11
We aren't saved by John. We ARE saved by Jesus - and it isn't water that saves us. That's why this is important. John's baptism was more or less ceremonial. It was important as a witness, yes, but it wasn't salvific.
John's baptism was a baptism of repentance :
Then went out to him Jerusalem, and all Judaea, and all the region round about Jordan,
And were baptized of him in Jordan, confessing their sins.
But when he saw many of the Pharisees and Sadducees come to his baptism, he said unto them, O generation of vipers, who hath warned you to flee from the wrath to come?
Bring forth therefore fruits meet for repentance:
Matthew 3:5-8
The Pharisees wanted to be baptized but John was saying they were missing the critical part of baptism - repentance. It was a ceremonial declaration of being clean. John prepared the way for Jesus.
If Jesus did it then you do it. That the devil in your ear saying you don't have to do it. DO IT ALL.!! If the book say He sang SONGS ALL DAY DO IT ALL.!!
Acts 1:1 The former treatise have I made, O Theophilus, of all that Jesus began both to do and teach,
ALL THAT JESUS BEGAN TO DO.!! Do it stop fighting it. You love the Scriptures.? Do it... You like fighting back and forth with the Scriptures, then do it. You not fight me you are fighting God.