"Christ died for all who would believe in him." Is this a clear statement? John Calvin said ambiguity is the fortress of heretics. Would saying something like Christ died for the elect whom the Father gave to Him based solely on His sovereign will be clearer?
The first statement is open ended, ambiguous and opens the door wide to semi-Pelagianism. I would turn it around and say, "All for whom Christ died will believe on Him."
The second statement is true but not exactly an alternative to the first because it doesn't address the matter of saving faith. The truth of the matter is nicely stated in Jesus' words:
John 10:26-27 (ASV) 26 "But ye believe not, because ye are not of my sheep. 27 My sheep hear my voice, and I know them, and they follow me:"
It is not uncommon for man to turn this text on its head (which the first statement is guilty of) and say, "You are not of my sheep, because you don't believe."
Sorry for the brevity.