Baptism "with" the Holy Ghost and Baptism "by" the Spirit.
The 12 were baptized "with" the Holy Spirit.
For John truly baptized with water; but ye shall be baptized with the Holy Ghost not many days hence. -Acts 1:5
But we on the other hand
For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit. -1 Corinthians 12:13
That's talking about the church...the collective body of Christ.
The baptism of the spirit goes on long after Cornelius. Same one given at Pentecost, same one given to all those people with Cornelius, same one with Paul, same one throughout Acts and beyond. "One baptism"...have you even read that scripture?
Here we see baptized WITH the Holy Ghost - notice it's the same "as on us at the beginning.
Acts 11:15-16 - And as I began to speak, the Holy Ghost fell on them, as on us at the beginning. Then remembered I the word of the Lord, how that he said, John indeed baptized with water; but ye shall be baptized with the Holy Ghost.
Acts 13:9 - Then Saul, (who also is called Paul,) filled with the Holy Ghost, set his eyes on him.
Acts 15:8 - And God, which knoweth the hearts, bare them witness, giving them the Holy Ghost,
even as He did unto us;
This shows how some didn't even know about the Baptism of the Holy Ghost. Same baptism as Acts 2.
Acts 19 - 1And it came to pass, that, while Apollos was at Corinth, Paul having passed through the upper coasts came to Ephesus: and finding certain disciples, 2He said unto them,
Have ye received the Holy Ghost since ye believed? And they said unto him, We have not so much as heard whether there be any Holy Ghost. 3And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism. 4Then 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. 5When they heard this, they were baptized in the name of the Lord Jesus. 6And when Paul had laid his hands upon them, the Holy Ghost came on them; and they spake with tongues, and prophesied. 7And all the men were about twelve. 8And he went into the synagogue, and spake boldly for the space of three months, disputing and persuading the things concerning the kingdom of God.