The only difference was after the dog was killed, the farmer would deliver the pet to the owner with an apology. The apology was sincere, but it was kind of "Sorry I shot your dog, it was chasing the cows." The owner usually replied, "Well, I guess it had to be done. Thanks for bringing Rover back to me."
but I thought some dogs chased cows! Don’t sheep dogs do that to “round them up”?
In the Texas hill country the sheep/goat ranchers routinely go through town and poison all the stray dogs.
In the State of Kansas, dogs bothering livestock may be shot, no questions asked. That’s the law.