Where do these equivalent terms mention "water baptism?" John 3:5 says
born of water and the Spirit, not born of baptism. 1 Corinthians 12:13 says - For by
one Spirit we were all baptized into
one body--whether Jews or Greeks, whether slaves or free--and have all been made to
drink into one Spirit. Plain ordinary H20 has no power to place you into the body of Christ. This is a clear reference to
Spirit baptism, not water baptism.
Now compare with John 4:10 - If you knew the gift of God, and who it is who says to you, 'Give Me a
drink,' you would have asked Him, and He would have given you
living water. John 4:14 - but whoever
drinks of the
water that I shall give him will
never thirst. But the
water that I shall give him will become in him a
fountain of water springing up into everlasting life. Jesus connects this living water here with everlasting life. Living water is
not water baptism.
The washing of regeneration is a reference to spiritual washing or purification of the soul, accomplished by the Holy Spirit through the Word of God at the moment of salvation. Washing of water by the Word (Ephesians 5:26). Having been born again, not of corruptible seed but incorruptible, through the Word of God which lives and abides forever (1 Peter 1:23).
See how this all fits together?