Now that you understand that part, I'll admit I was oversimplifying slightly.

Vasoconstriction is happening all around the region of injury. It's not really affecting just that big artery in the wrist. It's affecting all the little arteries and arterioles near the wound.

Constriction proximal to the injury will reduce blood flow to it, no matter how far proximal. It will work even if it's just happening in the arterioles a few centimeters or millimeters away from the injury.

That's what they mean by "at the injury site."