Cornelius and friends were a unique bunch. They were converted at a time in which the church was in transition. The circumcision group had a great degree of control and this affected Peter. So God did a work, to include Gentiles in the fold, by proving that Gentiles could be saved, when He gave them the Holy Spirit and the gift of tongues. Notice that Peter made sure that they were baptized who had been saved. "Can anyone forbid water, that these should not be baptized, who have received the gift of the Holy Spirit even as we?"
Regardless of the church being in transition in the book of Acts, these Gentiles believed, received the gift of the Holy Spirit, spoke in tongues and were saved
BEFORE water baptism. (Acts 10:43-47)
The name is important, and it does indeed represent authority. But without the name you don't have the authority.
The name absolutely represents authority. Acts 4:7 - "And when they had placed them in the center, they began to inquire, "
By what power, or in what name, have you done this?" 8 Then Peter, filled with the Holy Spirit, said to them, "Rulers and elders of the people, 9 if we are on trial today for a benefit done to a sick man, as to how this man has been made well, 10 let it be known to all of you, and to all the people of Israel, that
by the name of Jesus Christ the Nazarene, whom you crucified, whom God raised from the dead
by this name this man stands here before you in good health."
Acts 4:17 - "But in order that it may not spread any further among the people, let us warn them to speak no more to any man
in this name. 18 And when they had summoned them, they commanded them not to speak or teach at all
in the name of Jesus.
Acts 5:40 - "And they took his advice; and after calling the apostles in, they flogged them and ordered them to speak no more
in the name of Jesus, and then released them."
Acts 8:12 - "But when they believed Philip preaching the good news about the kingdom of God and
the name of Jesus Christ, they were being baptized, men and women alike."
Acts 9:27 - "But Barnabas took hold of him and brought him to the apostles and described to them how he had seen the Lord on the road, and that He had talked to him, and how at Damascus he had spoken out boldly
in the name of Jesus. 28 And he was with them moving about freely in Jerusalem, speaking out boldly
in the name of the Lord."
Acts 16:18 - "And this did she many days. But Paul, being grieved, turned and said to the spirit, I command thee
in the name of Jesus Christ to come out of her. And he came out the same hour."
Act 4:10, Be it known unto you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom ye crucified, whom God raised from the dead, even by him doth this man stand here before you whole.
Act 4:11, This is the stone which was set at nought of you builders, which is become the head of the corner.
Act 4:12, Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.
Amen!
Therefore, if water baptism saves; and 1 Peter 3:20-21 (kjv) indicates that it might; then baptism in the name of Jesus Christ of Nazareth is the only thing that will do the trick.
1Pe 3:20, Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.
1Pe 3:21, The like figure whereunto even baptism doth also now save us (not the putting away of the filth of the flesh, but the answer of a good conscience toward God,) by the resurrection of Jesus Christ:
In 1 Peter 3:21, Peter tells us that baptism now saves you, yet when Peter uses this phrase
he continues in the same sentence to explain exactly what he means by it. He said that baptism now saves you-
not the removal of dirt from the flesh (that is, not as an outward, physical act which washes dirt from the body--that is not what saves you),
"but an appeal to God for a good conscience, through the resurrection of Jesus Christ" (that is, as an inward, spiritual transaction between God and the individual, a transaction that is
symbolized by the outward ceremony of water baptism).
Just as the eight people in the ark were "saved THROUGH water" as they were IN THE ARK. They were not literally saved "by" the water. Hebrews 11:7 is clear on this point (..built an
ARK for the
SAVING of his household). The context reveals that ONLY the righteous (Noah and his family) were DRY and therefore SAFE in the ARK. In contrast, only the wicked in Noah's day came in contact with the water and they all perished.
Actually, in Colossians 3:17 it says that whatsoever you are going to do in word or deed, it needs to be done in the name of Jesus Christ. And I would say that baptizing people is something that we do in word and deed.
Baptizing people is something that we do in the name/authority of Jesus Christ "after" we believe the gospel and obtain salvation. (Acts 10:43-47; 15:7-9; Romans 1:16)