You state that I left out verse 13, but in fact I stated both 12 and 13. Please reread if necessary.
I originally quoted from the NKJV. The ESV makes it more clear, and I am now including verse 11 that gives more understanding to the Apostle's statement.
He came to his own, and his own people did not receive him. But to all who did receive him, who believed in his name, he gave the right to become children of God, who were born, not of blood nor of the will of the flesh nor of the will of man, but
of God.
(Joh 1:11-13, ESV)
One must first receive and believe in His Name, then, and only then does the Lord give them the right
to become a children of God.
The Greek word become is γίνομαι (ginomai).
Thayer defines it as "to come into existence, begin to be, receive being, to become, be made"
Word Study defines it as: "meaning to begin to be, that is, to come into existence or into any state"
Mounce defines it as: "to come into existence; to be created, exist by creation, to be born"
The Apostle's statement is very clear. One must receive and believe in His name for the Lord to give them the right to become-come into existence, to be created, to be born-children of God.
Chronologically, to become-come into existence, to be created, to be born-children of God, one must receive and believe in His name first, and then verse 13 defines it is God who births the receiving and believing as His child.
Please read all of the author's statement, for he states things quite clear. The bold words below in the author's comment help to define what he meant.
"He was teaching the necessity of the application of His atoning work.
Only when sin is dealt with in the life of the individual can that person experience life and move into the sphere of God’s holy Kingdom. Jesus made it clear that the soul cleansing benefits of His atoning work are given only to those who “believe” in Him."
To answer your questions.
Atonement means to make reconciliation, which our Lord did on a cross for all men. Now, it is up to each person to "receive the abundance of grace and the free gift of righteousness" (ref, Rom 5:17)", for we know "him who for a little while was made lower than the angels, namely Jesus, crowned with glory and honor because of the suffering of death, so that by the grace of God he might taste death for everyone. (ref, Heb 2:9)"
- All we like sheep have gone astray; we have turned—every one—to his own way; and the LORD has laid on him the iniquity of us all. (Isa 53:6)
- For the love of Christ controls us, because we have concluded this: that one has died for all, therefore all have died; and he died for all, that those who live might no longer live for themselves but for him who for their sake died and was raised. (2Co 5:14-15)
- All this is from God, who through Christ reconciled us to himself and gave us the ministry of reconciliation; that is, in Christ God was reconciling the world to himself, not counting their trespasses against them, and entrusting to us the message of reconciliation. Therefore, we are ambassadors for Christ, God making his appeal through us. We implore you on behalf of Christ, be reconciled to God. (2Co 5:18-20)
- But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: for all have sinned and fall short of the glory of God, and are justified by his grace as a gift, through the redemption that is in Christ Jesus, whom God put forward as a propitiation by his blood, to be received by faith. This was to show God's righteousness, because in his divine forbearance he had passed over former sins. (Rom 3:21-25)
The truths above are why it is necessary to receive and believe in the name of the Lord, for it is by the abundance of God's grace that He justified all men, everyone in the death of His Son, and this free gift of righteousness is only received by faith.
For “everyone who calls on the name of the Lord will be saved.” How then will they call on him in whom they have not believed? And how are they to believe in him of whom they have never heard? And how are they to hear without someone preaching? And how are they to preach unless they are sent? As it is written, “How beautiful are the feet of those who preach the good news!” But they have not all obeyed the gospel. For Isaiah says, “Lord, who has believed what he has heard from us?” So faith comes from hearing, and hearing through the word of Christ. (Rom 10:13-17)
The quote above disproves your understanding that I "believe because your sins were atoned for at the cross.". The driving factor in my belief is hearing that my sins were atoned for on the cross by God's Son, Jesus Christ. There is a difference. You are stating because my sins were atoned, I believe. When in fact the sins of everyone were atoned by the Lord's death, and one must believe it to receive it.
This is God's message through His own Son, “For God so loved the world, that he gave (delivered up) his only Son, that whoever believes in him should not perish but have eternal life. For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him." (Joh 3:16-17)
If the above message is true, and your understanding is true, then why don't all men believe for God delivered up His Son for the sins of us all, not to condemn the world but that the world might be saved through Him?
You can't have it both ways.
Faith comes by hearing the message of God's salvation through His Son preached.
Your misunderstanding of God's grace is causing you to deny that very grace of God towards all men.
For the grace of God that brings salvation has appeared to all men, teaching us that, denying ungodliness and worldly lusts, we should live soberly, righteously, and godly in the present age, looking for the blessed hope and glorious appearing of our great God and Savior Jesus Christ, who gave Himself for us, that He might redeem us from every lawless deed and purify for Himself His own special people, zealous for good works.
(Tit 2:11-14)
For this is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth. For there is one God and one Mediator between God and men, the Man Christ Jesus, who gave Himself a ransom for all, to be testified in due time,
(1Ti 2:3-6)
Truly, these times of ignorance God overlooked, but now commands all men everywhere to repent, because He has appointed a day on which He will judge the world in righteousness by the Man whom He has ordained. He has given assurance of this to all by raising Him from the dead."
(Act 17:30-31)
God Bless