Lichtenfeld's essay on vigilante movies covers a lot of ground, and much of it accurately. I've heard it said that a conservative is a liberal who has been mugged, which explains a number of the vigilante movies.
However, I don't believe Eastwood's Dirty Harry series belongs in the category of vigilante movies. Inspector Callahan does not meet the criteria Lichtenfeld established for a vigilante.
First, he's a cop. A damned good one, but an old-school cop. He carries a badge and mostly plays by the rules. He might bend the rules, but in the entire series he only broke them once - when, in Sudden Impact, he knew it was Sondra Locke's character who had murdered all those present or participating in her and her sister's rapes except the Sheriff's son, whose guilt led to a failed suicide that left him a vegetable. He had a chance to pin the murders on a psycho villain who had murdered the Sheriff, and took it. I'll return to this point.
Second, think back over the series. Can you come up with a single instance where Harry killed somebody who was not trying to kill him, mostly by shooting at him but once by ordering his assassination (the Hal Holbrook character in Magnum Force)? I can't. The definition of a vigilante is someone who kills for reasons of vengeance, whether or not he's under attack at the moment. Harry does not fit that definition.
Third, as I just mentioned, vigilantes go hunting for them what wronged them, with the intention of killing them, period. That's not Harry's style. The closest he got to that in the series was in the first movie, where he hunted down the Scorpio Killer and in the final scene of the movie, blew him away when he stupidly went for his gun on the dock when harry had the drop on him. And even there, even though the killer had hijacked a busload of school kids, Harry gave the killer a chance to surrender. He did not kill him until he had no choice but to do so. That's not vigilante behavior. Vigilantes look for excuses; or else don't even try to make any because they view their opponents as not being human.
Finally, the motto of the vigilante is, "Get even." Harry's motto is, "See that justice is done." He has limited confidence in the courts to see to that; after seeing the Scorpio Killer walk because the judicial system took exception to his searching the guy's quarters without a warrant, for instance, it's easy to see why he's cynical about judges and prosecutors and desk jockeys in general.
That is why he pinned all of Locke's murders on the psycho who had raped her and tried to kill both her and Harry in Sudden Impact. Justice, from his point of view, is the rapists and their enabler Ray Parkins (played by Audrie Neenan) ending up dead for what they did. They won't rape any more innocent or naive girls.
It's why he tossed his badge into the water at the end of Dirty Harry; he felt he had crossed a line when he went hunting for the Scorpio Killer. Perhaps he did intend to bring him in alive if he could, but there may have been lurking in the back of his mind the thought that the Scorpio Killer deserved to die for his crimes. To Harry, popping a bunch of mobsters who have been trying to kill him with .45s and Uzis in a parking garage even though they've expended most or all of their ammo is self-defense; they were trying to kill him, after all, and who is there to say how it went down but him? But blowing away the Scorpio Killer when he had the drop on him, even though the Scorpio Killer went for his gun and would have killed Harry if he could - that's shooting a sitting duck. That's murder, in his mind. And that's why he tossed his badge; he felt he no longer had the right to wear it.
(I don't think there were any witnesses to what went down on the dock. The kids and the bus driver like as not bugged out the first chance they got after Harry and the Scorpio Killer ran into the gravel crusher building. There is no one to say how it went down but Harry. A shooting board, I think, would have to call it a righteous shoot, self-defense, since the loon did go for his gun. But Harry knows, and he thinks he stepped over the line. This point was, of course, totally ignored when the movie became a smash and a sequel was planned. But that scene still says a great deal about Harry Callahan.)
No, I do not think Inspector Harry Callahan qualifies as a vigilante. A dinosaur, sure. An anachronism, possibly. But somebody out for vengeance for a wrong done to them personally to to their family? Not even close. Sorry. He's not a vigilante.