You're assuming water baptism when elsewhere Paul says we are baptized in His death. And John the Baptist said that Jesus doesn't baptize with water. So if there is only one baptism, as Paul wrote, then I'm pretty sure it's Jesus' baptism with the Spirit and fire, and not water like John's.
I can tell you really haven’t studied Holy Spirit baptism. In scriptures it only happened twice and each time it was for a special purpose and we are told out right what it was.
1) The apostles
Acts 1:5
for John baptized with water, but you will be baptized with the Holy Spirit not many days from now."
Acts 2:4
And they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit was giving them utterance.
2) Cornelius
Acts 10:44
While Peter was still speaking these words, the Holy Spirit fell upon all those who were listening to the message. 45 All the circumcised believers who came with Peter were amazed, because the gift of the Holy Spirit had been poured out on the Gentiles also. 46 For they were hearing them speaking with tongues and exalting God.
In chapter 11 Peter is describing what happened and if you will notice he says the HS fell on them just as it did the apostles “
at the beginning”
Acts 11:15
And as I began to speak, the Holy Spirit fell upon them just as He did upon us at the beginning. 16 And I remembered the word of the Lord, how He used to say, 'John baptized with water, but you will be baptized with the Holy Spirit.'
So we have two special situations where the bible tells us people were baptized by the Holy Spirit and each time it was directly from God and evident by speaking in tongues.
Yes there is one baptism and it's water. You’re using 1 Cor. 12:13 and trying to make baptism Holy Spirit baptism. We don’t even have to leave I Cor. to see that doesn’t work. In chapter 1 when Paul says 1:16,
now I did baptize also the household of Stephanas; it’s clear this is water baptism performed by Paul. I can’t imagine someone claiming Paul performs Holy Spirit baptism.
Still in 1 Cor. if we look at chapter 10 we see Paul describing the baptism of Moses.
10:1
For I do not want you to be unaware, brethren, that our fathers were all under the cloud and all passed through the sea; 2 and all were baptized into Moses in the cloud and in the sea;
Twice he says “under the cloud and in the sea”. He is clearly talking about water baptisms.
Then in chapter 12 when Paul says baptism it’s not water but Holy Spirit? What Paul means is that when we are water baptized “
in the name of Jesus” the Holy Spirit adds us to the body. For example, in Acts 2: 41
So then, those who had received his word were baptized; and that day there were added about three thousand souls.
They were baptized then the Spirit added them.
If you would consider 2 Kings 5 where Naaman was cleansed of his leprosy. When he dipped 7 times as told, did the water cleanse his leprosy or God?
Lastly, if you believe we are baptized by the Holy Spirit upon belief, could you please explain why Paul would ask,
"Did you receive the Holy Spirit when you believed?"
If the Holy Spirit baptizes us into the body upon belief, why would Paul ask
"Into what then were you baptized?"