You overlook the fact that the context indicates that the words : born of water and of the Spirit" is exactly the same thing as being "born of the Spirit":
"Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. Marvel not that I said unto thee, Ye must be born again. The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit" (Jn.3:5-8).
As I indicated earlier, this passage only speaks of one source of birth and not two. And the following words of the Lord Jesus shows us how one receives life when born again:
"It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, they are spirit, and they are life" (Jn.6:63).
And this matches perfectly with what Peter says here about how a person is "born again":
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Being born again, not of corruptible seed, but of incorruptible, by the word of God...And this is the word which by the gospel is preached unto you" (1 Pet.1:23,25).
EXACTLY! There is only one thing which results in the new birth, and that is the gospel which comes in the power of the Holy Spirit. And the gospel brings salvation to all who believe it:
" For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth" (Ro.1:16).
Simple as can be!