Calvinism teaches that not all people are the "elect", and that only the "elect" will be saved. This sounds good, until you see it contradicts other Scriptures. So, we develop a theology to encompass all the passages, rather than ignore some that differ with our view, or rather than misinterpret it to fit our view.
The elect will be saved, but not because they were chosen and others were not. The elect are those who believe that Jesus' death was for them. We KNOW that because Jesus died for the sins of the entire world, not just some: "He is the expiation not for our sins only, but also the sins of the entire world"(1 Jn.2:2), the same "all" of Paul in 1 Tim.2:4.
You said John 3:16 "literally says: 'that all the one believing shall not be destroyed but shall have eternal life". Actually, the literal text says, “God so loved the world, that He gave His Son, that whoever believes in Him will not perish, but have everlasting life.” WHOEVER.
Two of the points where Calvinism is wrong:
1) God didn’t just love the elect. He didn’t just love those who would DO something,"believe and I will then save you", because salvation is by grace alone on account of Christ. He loved the WORLD. He is not willing that ANY perish, but that ALL come to know Him. He saved us, rendered atonement on our behalf, but we must believe.
He died for all sins. All debt was paid in full. The Father was reconciled. We have forgiveness. Not some of us. ALL of us. Every person in the kosmos. But some will not believe.
2) Election is misdefined. You said “John 3:16 does not identify the nature of those who believe, but rather that all who believe will be saved.” No, it does identify that their nature is in the category of all, not some. All.
When a person believes that sufficient death of the sinless Son of God was FOR ME, he is called one of the elect. He is a believer. I think you can see that, because if we take your same quote, and apply it on "what does'elect' mean?", your quote reads:
"‘Elect’ in scripture is not the nature of some and not others, but rather, all who believe are elect."
See? Christianity is not for some, but forgiveness offered to everyone. To the entire world. When you believe, you are called the elect, believers, children of God, forgiven, redeemed, etc.
Election is not God picking some and not others. He picked ALL, and died for the sins of the "entire world."
This makes Christianity different than all other religions in the world. We offer a relationship with God to everyone, not just some. God's love is offered without that person doing anything of merit. We offer salvation and eternal life NOT because THEY have done anything, but because JESUS did everything. Forgiveness is not because "Well, I'm one of the elect", but because sinless Christ paid the just requirement as our Substitute.
This fits the actual texts of the Bible. We don't have to read something invented 1500 years later back into it. Calvin wasn't wrong on everything, was right on alot. But like we see on Travesty Broadcasting Network, when someone's wrong in one area, it really can affect their teaching badly. We need to be open to give up false teaching when it doesn't jive with the inerrant, inspired Word of God (2 Tim.3:16).
Blessings!
You have also ignored the breath of the term 'kosmos' (world) in John's gospel. The term can legitimately mean all the nations.
To maintain your view of univesal atonement you must also handle John 6:35-47, letting the text say what it says... I challenge you to do so.
Blessings to you as well.