Those believing on Him already have eternal life:
"What must I do to be saved? Believe on the Lord Jesus Christ because you are already saved".
Wait. It doesn't say that, does it? One of the big mistakes we make in theology is to have verses mean something that fits our theology - when the purpose of the verse was not really to prove that point. The verses you listed do explicitly say that a saved person is one who believes. That their purpose to us was to show the order of salvation is very weak in my opinion.
Belief on Christ is the biggest indicator one can have that one has been born of God.
Again. I agree 100 percent. It proves nothing about the order, although that may very well be the correct assessment of the order. What make this a little more difficult to understand is that "belief" when meaning saving faith is more that bare belief. Part of saving faith involves trust and a commitment or a closing with Christ as Lord and Savior. That involves your conscious mind and will. That is something you do. But on the other hand, theology aside, how do you decide that something is true? How do you become convinced? Is it enough that your rational mind evaluate the warnings and propositions of the gospel and then you really can will that this be true? Or is that a supernatural work. I say it is supernatural. Where I think strict Calvinists have a slight error is that I believe you can be convinced supernaturally that this is true, become aware of your condition and danger, and yet there is still a chance that you can evaluate this and love the world or your sin, or be too much interested in the flesh - and decide not to come to Christ for salvation. I could be wrong but that to me is the only way to take into account all the verses that show salvation as all God's initiative and yet have the guilt of refusal be truly on the one who refuses.
Now, a strict Calvinist will simply frame this as the rational mind that evaluates the propositions of the gospel and then comes to Christ has done so because he was just born again. And I have no problem with that either. By the way, Thomas Schreiner, who you quoted, was at our church a few years ago. If you ever get a chance to hear him do so.
By the way, Martin's post and assessment of all this was very good and I disagree with you in that I am the Calminian, not him.