No, the sequence isn't correct. Belief and re-birth are simultaneous. Eph 1:4 says believers were chosen before time but the moment of rebirth occurs with faith.
But John wrote: “...but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.
And how does this contradict Calvinism? Calvin taught that belief i.e. faith brings life.
Do you live, then believe? That is Calvin.
This is false.
Calvin taught we are DEAD in sin, and need to be born again in order to receive belief. Scripture teaches we are alienated from God (think about the Prodigal Son) and doomed to eternal death unless we believe the promise of God in Christ Jesus.