When Davis says, "If I'd been there sooner, he might still be alive," Davis wasn't just putting on airs. If he'd finished the job sooner, he wouldn't have had to murder the cop who saw him at the crime scene.
Scorpio tells Harry he's going to kill the girl anyway after being paid the ransom, even though Harry reached all the locations within the fourth phone ring. But, Scorpio also told him he couldn't talk to anyone on the way, and that he'd be watched at some points but wouldn't know when. Harry spoke to Alice at the Cross, whom he mistook for Scorpio, which Scorpio almost certainly saw, so technically, Harry did break one of Scorpio's 'rules'.
In The Enforcer, Harry objects to being required to obverse a demonstration of the M72 LAWS rocket launcher on the basis that he is already familiar with it. During the demonstration, Kate positions herself behind the instructor, and Harry pulls her out of the way just before the instructor fires the LAWS.
The fallout of Briggs' conspiracy in Magnum Force. Even if contained, Harry's killing of four officers, albeit dirty cops who had formed a vigilante death squad, would not go unnoticed (effectively a Xanatos Gambit by Briggs).
Okay, at the end of the first movie, Harry gets rid of his badge and is clearly not a cop anymore. However, in the sequel, he is still a cop working in the force, as though nothing had ever happened. Would someone like to explain that?
Magnum Force opens with Harry on loan to Stakeout, so maybe he was demoted as a result of his actions during the first film. Or for losing his badge.
He actually just threw his badge away as a sign, he didn't do this by the book and felt he wasn't a cop anymore.
Since when did Harry care about doing it by the book?
It's possible that he did intend to quite, but changed his mind before the second movie. As he remarks in that movie, the system isn't perfect but it's better than nothing. Maybe he figured he was in a better position to make a difference with a badge than as a civilian.