This is a good point. try to remedy the problem at the source which is the safety of the birds. But having had 20 birds killed last year by two very ellusive dogs. I will say, if the predator comes back shoot it or call the pound if it is your neighbors dog. foxes usually will kill and pack it off and cache it for later but not always. it sounds like a dog to me.Fox or Jack Russell is not relevant. Either could have done job, lack of consumption does not rule out fox, dog could have actually chased fox off before it had chance to consume kills. Pure speculation on that but possible although not likely.
Real issue is that either could have killed birds. Prevent next event and do not invest energy to deal with a single individual predator as cure to problem. Otherwise there will be a repeat by another individual.