Supplication is asking for Divine assistance. Prayer of this nature can be for the individual that is praying, or for assistance in another person's life. That sort of selfless prayer probably warrants a faster answer from the one who gave his own life.
It is a sacrifice of time, and a conscientious departing from one's own needs or desires, hence, it is a prayer of considerably real love for another.
I don't think the believer's prayer for salvation is altogether Biblical. Ministers and evangelists say to many that we should pray for Christ to come into our lives. I feel it is more of an act of faith when one decides to believe, to simply assert toward God a prayer of Gratitude for already being there. Now that one believes, he is righteous... confession is made unto salvation. Here is the exact quote from Romans.
"Man believes in the heart unto righteousness, and confession is made with the lips unto salvation."
The fits with the dictum that Christ put forth -- "Out of the abundance of the heart the mouth speaks."
Paul said in either Romans or Corinthians - "One cannot say that Christ is Lord unless it be by the spirit of God." So, to confess to the fact that Christ is God incarnate requires a moment called Metanoia, "repentance", which means to change one's mind. After the mind has genuinely changed, it is only logical that one's behaviors would be modified over time.
One can pray the "sinner's prayer" if it makes that person comfortable so that they can remember a time when they started their journey of faith. But I think God knows when a person really believes, and as for eternal salvation.... the only ingredient is belief.
The dude on the cross next to Christ never had time to change his behavior, for instance, but he changed his mind and Christ told him - "truly this day will thou be with me in paradise." But, as we live in Christ, we die to the deeds of immorality as a symptom of how we know that certain behaviors are merely illogical... Like sleeping bareback with a prostitute... or sleeping with one to begin with... health risks are high either way. If it is of the WORD, then it is of the LOGOS λόγος, meaning it is reasonable, as is God, who is the λόγος john1:1. God expects us to be loving, which is reasonable, and merely await the life prepared after all this πάθος, or, suffering is over.