First of all, I agree with you in disagreeing with that list. King Arthur's on there. While a great character, Arthur is decidedly not a 'superhero'. A better case could be made for Robin Hood, who is also on the list. He, after all, did fight injustice. And he had something of a secret identity (his real name was Robin of Locksley). But I would probably exclude him from the superhero list as well.
I think, however, that an exception could be added to your definition of superhero. I would say that agents for a specific government could be considered superheroes
if they in fact have super powers. I would point to WWII era superheroes who, if I'm not mistaken, spent a lot of time fighting Nazis (e.g. the original 'Human' Torch
http://www.fiveminute.net/comics/marvelcomics1.html). Also, I believe that Captain America was originally a super-soldier
Along the same lines, would the Scarlet Pimpernel be considered a superhero? The Scarlet Pimpernel was the secret identity of Baroness Orczy's character Percy Blakeney, who smuggle aristocrats out of revolutionary France. He was definitely fighting injustice and wasn't working on behalf of any official government organization. But his actions were directed primarily against France. I would be hard pressed to classify him as a superhero or not.