But the police need reasonable suspicion
before detaining you. You can't generate reasonable suspicion as a result of a detention attempt, as it has to exist before that point.
This is why the cop will need to claim in his report that the running individual appeared to match the description of the suspect, however vague it may have been.
"Be advised that the suspect is... human, with a height, weight, skin color, and gender."
"Roger that, dispatch, I see someone matching that description now. Moving to intercept."
In your example, the suspicious factors are "wielding/brandishing a weapon in the absence of a known threat" and "possible suspect or witness to a violent crime". Since the person could be a frightened witness who pulled the knife in self-defense, it would be inappropriate to shoot him in the back 0.5 seconds after ordering him to drop the knife.
If the very same person did the same thing in the woods, near a dead deer, the suspicious factors would be "this is not deer season", "there was no gunshot", and "he isn't wearing safety orange". There are plenty of factors there to justify an investigatory detention beyond the direction and magnitude of a person's motion vector. But they must be relevant to a specific crime.
In the street example, the suspected crime is manslaughter. In the woods example, it is poaching. Without the body, the cop would have to fall back on, perhaps, city ordinances regarding the blade length of knives carried in public. Reasonable suspicion requires that the target be suspected of something specific.
If a cop chases down someone who ran from him upon spotting him, what is the suspected crime? As far as I know (not a lawyer), it is only a crime to flee from police custody if you are already suspected of a different crime, for which they already have reasonable suspicion to detain you. You can't be reasonably arrested solely for resisting arrest, because the arrest cannot occur prior to suspicion for an arrestable offense.
That's why you sometimes see reasonable articulable suspicion. If the cop cannot say why he chose to arrest you at the time of the arrest, it might be a bogus arrest. "Being a suspicious person" is not by itself a crime.
As for respect, the public owes the police respect in the same way that they owe restaurant waiters a 15% gratuity. If the waiter pees in your soup, you shouldn't leave a tip. If a cop fails to uphold the public's mandate for policing, he shouldn't expect any respect. That obligation flows only one way. If the public does not respect the police of a particular polity, it is because those particular police are providing them with poor service. It is not our responsibility to "save their tip".
In the waiter analogy, the customer does not leave a tip, so the next time he visits the restaurant, the waiter pees in his soup AND suckerpunches him in the face, then complains loudly when the customer not only fails to leave a tip, but also declines to pay for the soup.