With such a powerful draw out of position, I'd try to get the money in on the flop. You are ahead of a
dry set, and the villain could easily have a much weaker hand, perhaps even one he would fold if you show some aggression. In the actual hand, you were a
small favorite on the flop.
On the turn, why not push? You don't have to discount your diamond outs very much against one opponent. I think the chance he has a higher diamond draw is about 10%. You are likely to be
about even against some strong hands, but not much of a favorite against a weak hand. Give the weak hands a chance to fold, and don't give a free card to what might be a pure draw.