In John 3:5, Jesus said "born of water and the Spirit" and NOT born of baptism and the Spirit. Jesus mentions "living water" in John 4:10, 14; 7:37-39 and in John 7:38-39, we read - "He who believes in Me, as the Scripture has said, out of his heart will flow rivers of living water. But this He spoke concerning the Spirit. The Holy Spirit is the source of living water and spiritual cleansing. If "water" is arbitrarily defined as baptism, then we could just as justifiably say, "Out of his heart will flow rivers of living baptism" in John 7:38. If this sounds ridiculous, it is no more so than the idea that water baptism is the source or the means of becoming born again.
Also "water" is used in the Bible as an emblem of the word of God, and in such uses it is associated with cleansing or washing. (John 15:3; Ephesians 5:26) When we are born again, the Holy Spirit begets new life, so that we are said to become "partakers of the divine nature." (2 Peter 1:4) The new birth is brought to pass through "incorruptible seed, by the word of God, which lives and abides forever" (I Peter 1:23) and the Holy Spirit accomplishes the miracle of regeneration.
In regards to water baptism in John 3:22, by this time Jesus' conversation with Nicodemus is over with and He has moved on. John 3:22 - After these things Jesus and His disciples came into the land of Judea, and there He was spending time with them and baptizing. Verse 22 is out of context with verse 5. What is still in context with verse 5 is verses 14-18.
John 3:14 - And as Moses lifted up the serpent in the wilderness, even so must the Son of Man be lifted up, 15 that whoever believes in Him should not perish but have eternal life. 16 For God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish but have everlasting life. 17 For God did not send His Son into the world to condemn the world, but that the world through Him might be saved. 18 He who believes in Him is not condemned; but he who does not believe is condemned already, because he has not believed in the name of the only begotten Son of God. *What happened to baptism?
John 3:18 - He who believes in Him is not condemned; but he who (is not water baptized? - NO) does not believe is condemned already, because he has not (been water baptized? - NO) because he has not believed in the name of the only begotten Son of God.