Jesus was NOT born of the written word. Nor are we who claim to be "Born Again". Therefore, I believe Peter was referring to the living Word, not the written one.
Peter tells us exactly the identity of of the "word" by which people are born again:
"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).
James says the same thing:
"He chose to give us birth through the word of truth, that we might be a kind of firstfruits of all he created" (Jas.1:18).
Of course being born again is the same thing as being "born of God" and we can see that first century Jewish believers were born of God when they believed:
"He came unto his own, and his own received him not. But as many as received him, to them gave he power to become the children of God, even to them that believe on his name: Who were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God" (Jn.1:11-13).
Those who believe that Jesus is the Christ, the Son of God are born of God:
"Whosoever believeth that Jesus is the Christ is born of God: and every one that loveth him that begat loveth him also that is begotten of him. By this we know that we love the children of God, when we love God, and keep his commandments. For this is the love of God, that we keep his commandments: and his commandments are not grievous. For whatsoever is born of God overcometh the world: and this is the victory that overcometh the world, even our faith. Who is he that overcometh the world, but he that believeth that Jesus is the Son of God?" (1 Jn.5:1-5).
The application of new birth was "Christened" in John 20:21,22 per John 14:17 KJV: "Then said Jesus to them again, Peace be unto you: as my Father hath sent me, even so send I you. And when he had said this, he breathed on them, and saith unto them, Receive ye the Holy Ghost [indwelling]." John 20:21-22 (KJV)
The correct translation of John 20:22 is as follows:
"And with that he breathed on them and said, 'Receive holy spirit'" (Jn.20:22).
The Lord Jesus said that the Holy Spirit would not be sent until He departed to be with the Father and He had not yet departed to be with the Father when He said what He said at John 20:22:
"But now I go my way to him that sent me; and none of you asketh me, Whither goest thou? But because I have said these things unto you, sorrow hath filled your heart. Nevertheless I tell you the truth; It is expedient for you that I go away: for if I go not away, the Comforter will not come unto you; but if I depart, I will send him unto you" (Jn.16:5-7).