Baptists have never taught that baptism is necessary for salvation. You must be thinking of the Catholic Church or Church of Christ (denomination).
And, no, that is the correct interpretation of baptism. People are SAVED FIRST then baptized. So the baptism with the Holy Spirit (resulting in the the New Birth) comes first, then Christian baptism by immersion in water.
Act 8:14 Now when the apostles which were at Jerusalem heard that Samaria had received the word of God, they sent unto them Peter and John:
Act 8:15 Who, when they were come down, prayed for them, that they might receive the Holy Ghost:
Act 8:16 (For as yet he was fallen upon none of them: only they were baptized in the name of the Lord Jesus.)
Act 8:17 Then laid they their hands on them, and they received the Holy Ghost.
Act 19:2 He said unto them, Have ye received the Holy Ghost since ye believed? And they said unto him, We have not so much as heard whether there be any Holy Ghost.
Act 19:3 And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism.
Act 19:4 Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus.
Act 19:5 When they heard this, they were baptized in the name of the Lord Jesus.
Act 19:6 And when Paul had laid his hands upon them, the Holy Ghost came on them; and they spake with tongues, and prophesied.
Act 10:44 While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.
Act 10:45 And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.
Act 10:46 For they heard them speak with tongues, and magnify God. Then answered Peter,
Act 10:47 Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
Act 10:48 And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.
A person can receive the Spirit before baptism, or after baptism.