From the information presented in the article(s) and the demeanor of the shooter as well as his responses it is clear the guy is actually well reasoned and stable. If you can't see that from his desire to make amends and work it out with the dog owner, then you must not be reading the same story.  

If I was placed in the same position, I can't say for certain what I would do with a strange dog rushing my child in a seemingly unrestrained area. For all the guy knew, the dog could have been a stray that had taken up residence in the area.  

For the record, I own a dog, and she can be unpredictable, especially around strange adult men, she snarls and snaps inappropriately on occasion and I am keenly aware of it. As such, I make a point to keep her away from strangers, and in all honesty, if she got out and ran at someone and they shot her dead, I wouldn't hold the slightest grudge, it would have been my fault, even though she is generally a good dog and is a part of my family.