[CynicalBear]

>>does Paul ever use the term “spirit baptism”?<<

Peter did.

1 Corinthians 6:11 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.

[one Lord one faith one baptism]

you say Peter did, then quote Paul in Corinthians?

the verse you cite does not speak of “spirit baptism”, try again.
this verse teaches we are baptized by the Holy Spirit, thru the Church.