You seem to think that the second dog was killed in its owner's back yard. It's more likely that both were shot in "Wyatt Earp"'s yard, then one managed to wriggle back under the fence before assuming room temperature.
Hearing gunfire on your neighbor's property is NOT sufficient cause to start shooting at people. It would not be self-defense, and would be considered negligent homicide at the very least.
Given his previous threat(s), I find it more likely that he shot the second dog in its yard and then lied than I do that the wounded animal crawled back under the fence.
The fact is none of us know what really transpired. And I think the guy telling the story is full of it.