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Jericho "Action" Jackson is a Cowboy Cop (on TV Tropes). Captain Armbruster is Da Chief. Except neither of them really are. See the thing is, however poorly Action Jackson is executed, it seems to be trying to be something more interesting than just another rulebreaking-cop movie.

He's a loose cannon, but DAMMIT he's the best we have!

That's the Cowboy Cop. Furthermore:

Sure, our society may be built upon rules and procedures, but they usually make for bad television. Sometimes you have to bend the rules, rough up the suspects, moon your supervisors and shred the Constitution to get stuff done.

Jackson very much used to be that. As the two officers in that too-long cold open play him up as a bogeyman for criminals:

Kornblau: Yeah, some say he didn't even have a mother. That some researchers created him to be the first man to walk on the moon without a space suit. 

Lack: Mm-hmm. 

Kornblau: Others says his mother was molested by Bigfoot, and Jackson is their mutant offspring. 

Lack: They bring in Jackson when they want to re-educated some young ne'er-do-well such as yourself, Albert. 

Kornblau: Yeah, I remember one kid got re-educated so bad, his testicle climbed back up into his belly. Wouldn't come out. 

Lack: They called it medical miracle. 

Kornblau: Yeah. Another kid, handcuffed to a chair, gnawed his own hand off like a trapped skunk, or wolverine, or something.

But, when Albert, that purse snatcher from the cold open, tries to make a run for it--inside the police station, mind you--he ends up bumping into and making a mess of Jackson's desk. Jackson stands, says simply "Mellow out", and Albert passes out.

(Later Albert will happen upon Jackson out with Patrice and immediately run away. But there will be no third time to set a pattern and tie Albert, perhaps, to the climax of the film, which would have at least made the lone opening worth it.

But they are overselling him on purpose. Jackson lost his stripes two years ago. Still, in the present, Captain Armbruster calls him "dedicated". In the present, Jackson is not the rulebreaker anymore. And, despite Jackson having lost his stripes because of Dellaplane, Armbruster still thinks Jackson could represent the department at the Businessman's Association Dinner (or whatever that event is). And, even with one-on-one interaction with Dellaplane, Jackson manages to do okay. He sees the news report about Grantham dying in the yacht explosion and immediately sees something bigger is going on. He's a good cop. He's got a degree from Harvard. He's supposed to be better than a Cowboy Cop. But, for some reason, they gave him a Cowboy Cop backstory and took away his gun and expect him to be more Joe Friday (before act three) than Pep Streebek. Which could work if each beat of the investigation felt more earned, and any of it felt more cohesive.

Meanwhile, we don't hear from Armbruster for quite a while. And Jackson continues to "investigate" and behave himself pretty well. I complained yesterday about him backing off Dellaplane when he learns the other guy at the table is the president of the Auto Workers Alliance, but backing off in that moment is a good move for a good detective. Who doesn't have a Da Chief calling him out for all his shit, because there's not much shit to be called out on.

Da Chief, by the way:

This is the eternally put-upon superior of an organization police desperate to. They are always strict and by-the-book but can be comfortably relied upon to give a good Arson, Murder, and Lifesaving-style speech. You can always expect them to say that you have twenty-four hours or demand that you Turn In Your Badge, usually at the top of his formidable voice. They frequently worry that the mayor or district attorney will have his ass (and pension) for whatever destruction the Cowboy Cop caused.

But, for example, after Jackson chases down and jumps on the cab driven by a killer, and causes it to crash, the Captain doesn't already know about it, and doesn't get to yell at him over it because Jackson disappears as the Captain is on the phone finding out. Of course, had Armbruster yelled at him over that chase, it would have been a reasonable response anyway.

An actual Cowboy Cop would have had his first interaction with Dellaplane's wife not by chance meeting but having deliberately sought her out to mess with her husband. And he would have done the same to the mistress rather than her just happening to pick him out of a crowd while she sings. And he would deserve his Promotion, Not Punishment.
 
 
 
 
 
And I thought I was done with this movie. I mean, I'm pretty sure we don't see the captain again until everything is over. And, Jackson is going to commit some acts of violence but we're into act three now and that's just par for the course. But then he stole a police car (not that bad) and proceeded to endanger Sydney's life (or at least make her think he was endangering it) to get her to agree to help him (definitely bad). And, I don't think he even really needs her help at this point, but I guess they want Vanity to get more screentime.

But, worse still, later when Jackson drive's Dellaplane's new car into Dellaplane's mansion, he crashes right into the room where a) he knows Sydney is and b) doesn't know that she isn't right in the path of his car.

And, Captain Armbruster arrives at the mansion before Dellaplane's body is even cold and calls Jackson "Lieutenant" because, why not?