There is nothing in what you write about the Pentecost that contradicts that the Holy Spirit descended on a select group of people, who, lead by St. Peter, converted and baptized others.
May be your pastor taught you to read "born of water and the Holy Ghost" and mentally omit "water", but I read what is written. Water refers to baptism. Now, regarding the timing of this, Jesus speaks to Nicodemus in the future tense of entering the Kingdom of God, which places the baptism reference in the same time frame as salvation, following Christ's resurrection.
"There is nothing in what you write about the Pentecost that contradicts that the Holy Spirit descended on a select group of people, who, lead by St. Peter, converted and baptized others."
Those select group of people were more than the Apostles so there was no mediator, Apostles or church, between them and the Father's pouring out the Holy Spirit on them and they weren't baptized at that time. The new converts were but not the 120.
Water has many references in the New Testament like the "washing of water by the Word" in the Ephesians verse that has nothing to do with baptism. What baptism would Jesus be referring to, John's? The Pharisees had already rejected that and Jesus could forgive sins so John's was not necessary. could not look forward to His death because He had not announced it. Jesus wasn't baptizing anyone so what baptism was He referring to?