44 While Peter was yet speaking about these matters the holy spirit fell upon all those hearing the word.
45 And the faithful ones that had come with Peter who were among those circumcised were amazed, because the free gift of the holy spirit was being poured out also upon people of the nations.
46 For they heard them speaking with tongues and magnifying God. Then Peter responded:
47 Can anyone forbid water so that these might not be baptised who have received holy spirit even as we have?
48 With that he commanded them to be baptised in the name of Jesus Christ (Acts 10).
These men were commanded to be baptised in the name en tw onomati of Jesus Christ. Peter also says in Acts 2:38:
38 Repent and let each one of you be baptised in the name of Jesus Christ for forgiveness of your sins, and you will receive the free gift of the holy spirit.
39 For the promise is to you and to your children and to all those afar off, just as many as Jehovah our God may call to him' (Acts 2).
Being baptised in the name of Jesus is being baptised with the authority of Jesus. Peter was commanding a water baptism for Cornelius. He was already baptised in holy spirit for if not then he could not have received the gifts of the spirit. Gifts of the spirit are only given to angels (or born again humans with an angelic body).
The structure of baptism is explained by John, who says:
5 Who is he that conquers the world, but he who has faith that Jesus is the Son of God?
6 This is he that came by means of water and blood, Jesus Christ, not with the water only but with the water and the blood.
And the spirit is that which is bearing witness, because the spirit is truth.
7 For there are 3 witness bearers,
8 the spirit and the water and the blood, and the 3 are into the one thing (Literal 1 John 5:5-8).
34 Yet one of the soldiers jabbed his side with a spear and immediately blood and water came out (John 19).
49 Another man took a spear and pierced his side, and blood and water came out (Matt 27).
The blood is the human blood of the Christ who was the validation sacrifice for the 1AC. He ransomed Abraham. The water baptism is into the ICC and the spirit baptism is into the JAC - see intro28.
So the spirit the water and the blood are the baptisms into the JAC, the ICC and the 1AC. And the 3 are into one thing, which is the baptism into the ARC, the final sealing baptism into everlasting angelic life.
Acts 19 is quite hard to understand. It says...
1 In the course of events, while Apollos was in Corinth, Paul went through the inland parts and came down to Ephesus, and found some disciples;
2 and he said to them: Did you receive holy spirit when you became believers? They said to him: Why, we have never heard whether there is a holy spirit.
3 And he said: In what, then, were you baptized? They said: In John's baptism.
4 Paul said: John baptized with the baptism [in symbol] of repentance, telling the people to believe in the one coming after him, that is, in Jesus.
5 On hearing this, they got baptized into the name of the Lord Jesus.
6 And when Paul laid his hands upon them, the holy spirit came upon them, and they began speaking with tongues and prophesying.
7 All together, there were about 12 men (Acts 19).
The point is that Paul did not baptize them into he name of the Lord Jesus. We know this because when he laid his hands upon them they got the gifts of the spirit not a baptism. We also know this because Paul was not a 1NC saint so was unable to baptize into the 1NC. We also know this because the 1NC is a temple not made with hands. All those baptized into the 1NC are baptized by angels not men.
58 We heard him say, I will throw down this temple that was made with hands and in 3 days I will build another not made with hands (Mark 14).
One cannot be baptized by the angels in holy spirit until one knows what the holy spirit is.
In the case of the Corinthians in Acts 19, they had already been baptised in John's baptism, and so Paul did not baptise then in the name of Jesus, he baptised them into the name of Jesus.
Whereas the Samaritans in Acts 8 had been baptized into the name of Jesus but had not yet received the gifts of the spirit....
14 When the apostles in Jerusalem heard that Samaria had accepted the word of God, they dispatched Peter and John to them;
15 and these went down and prayed for them to get holy spirit.
16 For it had not yet fallen upon any one of them, but they had only been baptized into the name of the Lord Jesus.
17 Then they went laying their hands upon them, and they began to receive holy spirit (Acts 8).
Now we can understand why Peter said to the household of Cornelius: 'Can anyone forbid water so that these might not be baptised'. He said this because being Gentiles, like the Ethiopian Eunuch, they had not been baptised in water, so he commanded them to be baptised in water with the authority of Jesus Christ (i.e. in the name of Jesus Christ) because Jesus had authorized gifts of the spirit, so plainly he had also authorized water baptism which is the mechanism by which one joins the church.
The water baptism and the spirit baptism are nicely contrasted by Peter who says...
21 A few people, that is, eight souls, were carried safely through the water. That which corresponds to this is also now saving you, namely baptism, not the putting away of the filth of the flesh, but the request made to God for a good conscience, through the resurrection of Jesus Christ' (1 Peter 3).
The putting away of the filth of the flesh is the baptism of repentance, the baptism of John, it forgives the sins of the past. It does not result in a clean conscience. However the baptism into the name of Christ is a request for a good conscience, and it is no more than a request since no sons of God has authority over the holy spirit. God has authority over the holy spirit, his wife, but he has given all authority in heaven and on earth to Jesus (Matt 28:18).
Finally in Acts 2 we read...
38 Peter [said] to them: Repent, and let each one of you be baptized in the name of Jesus Christ for forgiveness of your sins, and you will receive the free gift of the holy spirit (Acts 2).
This was a baptism in water with the authority of Jesus Christ. For 3,000 souls were added to the congregation on that day and water baptism is how one is added to the church.
41 Therefore those who embraced his word heartily were baptized, and on that day about 3,000 souls were added (Acts 2).