If you choose to keep, as a companion, a beast that indiscriminately threatens anyone who approaches, including police officers lawfully discharging their duties, that's on you.
According to the article, we're talking about a 12 pound dog. I have a cat that weighs twice that much. A Jack Russell is usually a bit larger than that, so I'll be charitable, and assume the reporter made a mistake. Let's assume we're talking about a 25 pound dog -big enough that it *could* bite someone's shin or calf. Otherwise, we're talking an "ankle biter" and I really doubt the officer failed to wear shoes that day. Even a thick sock would've likely prevented any serious damage. Beyond that, biting a hand or face would require the officer to have done something to put those parts within reach.
Supposedly, the woman told the officer the dog doesn't bite. I'll argue that wasn't the best response: she should've had the dog trained to stop barking and go lie down or something once she told it that everything was okay.
Again, according to the links posted, the officer "gave the dog verbal commands" to "get back". I've owned dogs all my life, and "get back" has never been an order one of them understood. "Quiet now" or better yet "Sit!" would've made some sense. Again, I'll be charitable, and assume this particular officer had never seen a dog before, and had no idea what a marginally trained house pet would understand or be trained to follow. I'll also note that even if he did know, a well-trained dog would
be trained to disobey strangers.
We have the right to defend ourselves, our families, and our property. A pet dog is often the first line of defense: it starts to bark, and lets us know someone is in the driveway, ringing the bell, or crawling through the damned window. For most pets, that's the whole gig: bark 'til they're told not to, then back off and go lie down somewhere. A few of the more territorial breeds will continue to watch, ready to become aggressive again if they see something they think is wrong. Walk into a home and slap the owner of a German shepherd, and you just may lose your hand. That kind of defense is exactly what many people buy those dogs for.
As to the specifics of your post: very few house pets indiscriminately threaten anyone. They bark at strangers. They're far more likely to bark at adult men than anyone else, or at anyone dressed in a way that seems foreign to them. Again, they usually can be told to stop, and that's the end of it. I do not know why the owner of the dog in question didn't call the dog off, or if the dog wasn't trained to obey.
But, none of that changes the basic fact: a 300 pound man felt so frightened and threatened by a dog that he shot it -and not even in the head; it didn't die right away. That's a damn cowardly thing to do. Sorry, but it is. The man does not belong in any branch of law enforcement. He needs to pack up his marbles, and go find a nice desk job where cotton teddy bears are the most threatening thing around. Living, breathing people and pets are clearly far, far too much for him to handle.
Personally, I think they should compensate the woman for her dog, and fire him.