1 Cor 12:13 For by one Spirit we were all baptized into one body, whether Jews or Greeks, whether slaves or free, and we were all made to drink of one Spirit.
This verse would be a reference to the Corinthians having been born again per Christ's words of Jn 3:5.
Jn 3:5-------------Spirit+++++++++++++++water>>>>>>>>>>>>in the kingdom
1Cor12:13-------Spirit+++++++++++++++baptized>>>>>>>>>>in the body
Being 'in the kingdom' and 'in the body' are equivalent terms, they both represent a saved position. Since there is but only one way to be saved, then both verses must express the same idea. Spirit is equivalent to spirit and water is equivalent to baptized. So 1 Cor 12:13 is a reference to the water baptism of the new birth.
1 Cor 1:14, 16 "I thank God that I baptized none of you, but Crispus and Gaius.....And I baptized also the household of Stephanas..."
Paul baptized some of the Corinthians himself. Only God can administer baptism with the HS. In Acts 2 and Acts 10 God baptized the apostles and Cornelius directly without intervention of humans hands. Therefore Paul could only administer water baptism to the Corinthians. Paul, like all disciples, was under the great commission, Mt 28:19,20, Mk 16:15,16, which is disciple (human) administered water baptism, the same baptism Paul administered to some of the Corinthians himself. 1 Cor 1:14,16 is water baptism of the great commission. If 1 Cor 12;13 is some kind of spirit baptism that that is two baptisms and not one baptism per Eph 4:5.
1 Cor 6:11 "And such were some of you: but ye are washed, but ye are sanctified, but ye are justified in the name of the Lord Jesus, and by the Spirit of our God."
This washing would be a reference to the Corinthians having been water baptized and having those sins washed away by the blood of Christ. No verse in the NT says baptism with the Holy Spirit washes away sins.
Lastly, Jn 4:1,2 "When therefore the Lord knew how the Pharisees had heard that Jesus made and baptized more disciples than John, (Though Jesus himself baptized not, but his disciples,)"
Verse 1 says Jesus baptized yet verse two says He baptized not. This is not a contradiction for Jesus did not baptize anyone Himself personally, but He baptized by giving His authority to baptize to His disciples. Likewise the Holy Spirit baptizes the same way. The HS does not baptize anyone personally Himself, but baptized by giving authority through His word to the disciples to baptize. The HS baptized the Corinthians by giving Paul the authority to water baptism them.