Did the Elder Wand try to kill Harry?

If Harry needed to die, then the Elder Wand should have done it's best to follow Harry's intentions.

The excerpt below sounds like Harry wishing for his death to be quick. If the Elder wand is tuned in to Harry's intentions then the Elder Wand would be trying it's best to kill Harry.

Deathly Hallows chapter The Forest Again

Voldemort had raised his wand. His head was still tilted to one side, like a curious child, wondering what would happen if he proceeded. Harry looked back into the red eyes, and wanted it to happen now, quickly, while he could still stand, before he lost control, before he betrayed fear —

He saw the mouth move and a flash of green light, and every- thing was gone.

Does this mean that the Wand did try to kill Harry (to follow Harry's intent and Voldemort's too of course).