The winners of which sporting event are presented with the "Claret Jug"?

For those of us across the pond, Open might mean US Open. I always thought Derby was spelled darby in GB; I didn't know it was just pronounced that way.

It is not often called the 'British Open' at all. It is simply The Open - always has been, always will be. Just as the Derby is just that, not the British Derby. The originals do not need to have the word British in front of them!