I have read somewhere that losing a limb naturally (e.g. tearing, biting) stretches and closes up blood vessels in a way that cutting with a scalpel doesn't, and that natural limb loss has less bleeding than surgery/stabbing/Hollywood suggests.

I have no idea of the truth of this, but there are anecdotes of farmers losing limbs and walking back to their tractors, driving back to the farm, then driving themselves to hospital, and the like.