An animal is simply an animal. It is their relationship with a particular person that makes them a "pet" or "livestock".
You show me a dog roaming the neighborhood. I see only an animal, but you point the dog out to the owner, and they see their pet or family member.
Just as you wouldn't trust every person strolling down the street. They are strangers until you get to know them. Then they are an acquanitance, a friend, an enemy, or even family.
Someone's pet may be that dog in my backyard trying to eat my livestock, that dog is only a predator to me. Just as the pet owner, views my chickens "only" livestock.
Would I shoot a strange dog for attacking my chickens? You sure betcha. Would I shoot my own dog for attacking my chickens? No, "my" dog is more valuable to "me". That being said, would I shoot a dog for simply being on my property? No.
I'm sure this conflict between dogs and chickens has gone on as long as time itself. Nobody will ever be in total agreement, because each person is going to respond differently.
Sorry you lost your chickens. Since this is indeed your neighbors and you have to live near them, I would take a different approach. File a police report every time their dog takes a bird, and then ask for compensation. Make sure to ask HIGH for compensation $50 a bird is what I would do. If they have to start paying to replace these birds, they WILL do something about that dog themselves. And you won't be painted neighbor's family pet.