I came across a some more badges that may need to be corrected. I'm not sure what defines a "Combat Badge" and a "Quest Badge", so the following may be incorrect but these badges do seem to be categorized incorrectly based on their description.
The following Badges may be incorrectly categorized as "Combat Badges" when they should be "Quest Badges":
Knight of the Order of Fetch
The Big One
The following Badges may be incorrectly categorized as "Quest Badges" when they should be "Combat Badges":
I just received the badge Artful Dodger, but I received it after completing the quest "Twist of Fate". The badge description reads "Defeat the Master Criminal, Bill Pike." and the badge is categorized as a combat badge, but the badge is not received following combat but after completing the quest.
One more, if I may - I received the Badge "Yujin no Kunoichi" after defeating Master Ozawa -- before turning in the quest "The Silk One" - so possibly it should be categorized as a Combat Badge rather than a Quest badge.