The only way to prove a dog is a killer is to catch it in the act, or check it's teeth for hair right after the attack. The sheriff found some of the dogs that were killing our sheep by checking the teeth of suspected dogs for wool jammed between their teeth. Kind of like elementary CSI ..... lol. Of course in your case, too many hours have already elapsed, and presence of goat hair could simply be due to a dog coming across the goats that had already been killed and taking advantage of a free meal. Also, since the dog that you captured wanted to eat you too, checking its teeth probably wouldn't be real practical.
I have a sneaking suspicion that you have found the culprits. The next problem is proving it. If somehow you can prove it, and trace these critters back to their owner, you have a good case for recuperating some money for the damage. I know that even that would be poor compensation because I am assuming these goats were more pets than some kind of commercial enterprise. But some form of financial retribution would be better than nothing. And maybe it would teach the dog owner something about responsible pet ownership.
At any rate, it doesn't look good for proving the case that these dogs were responsible.
By the way, I agree with Cynthia ...... that dog looks pretty under-fed and neglected. He looks pretty "ribby" and boney in the hind-quarters. It may be a case of animal abuse.