There is only one Holy Spirit baptism, and that is the receiving of the Spirit.
Ephesians 4:5 says,
"One Lord, one faith, one baptism,"
The disciples were born again spiritually in their heart by the Spirit by their believing in Jesus and the Son (the 1st Comforter) lived inside of them (giving them eternal life). While their hearts were changed spiritually by the Holy Spirit (and the Spirit worked in their lives), the Spirit did not live in them yet until Jesus breathed the Spirit on to them (Thereby giving them another Comforter). Pentecost was the baptism of the Spirit via for 3,000 thousand Messianic Jews (and not the Jews on a national level). The conversion of Cornelius (and his acceptance of the gospel) was the baptism of the Spirit to show that Gentiles are now accepted into God's plan of salvation, as well.
When men of God (in the early church) accepted the gospel, many times they received the Spirit, and this would include the gifts. In one instance, the Spirit did not come upon certain individuals, and so they sent Peter and John so that they might pray for them to receive the Spirit. When that happened, they received the Spirit with power. Simon the sorcerer had seen this as gaining power, and he thought he could buy the Spirit (Which showed his heart was not right with God). This proves that with the receiving of the Spirit, comes with the power of the Spirit. So there is not two Spirit baptisms (See Acts of the Apostles 8:9-24). For Cornelius and his household had spoke in tongues when the Spirit first came upon them. Speaking in tongues is one of the many gifts of the Spirit. Cornelius to our knowledge never received the gospel before and or the Spirit before Peter came to his house (See Acts of the Apostles 10:44-46).