“All this chatter just increases the idea that these encounters are avoidable and law enforcement is at fault,” said Jeff Roorda of the St. Louis Police Officers’ Association, who said officers already thought about ways to avoid confrontations.

The typical police cadet receives about 58 hours of training on how to use a gun and 49 hours on defensive tactics, according to a recent survey by Mr. Wexler’s group. By comparison, cadets spend just eight hours learning to calm situations before force is needed, a technique called de-escalation.

“Everything now is: You get there, you see a guy with a knife, you resolve it,” said Mr. Wexler, a former senior Boston police official. In many situations, he said, officers who find themselves 21 feet from a suspect can simply take a step backward to buy themselves time and safety.

Mr. Tueller’s article never proposed a bright line between a shooting that was justified and one that was not. In a telephone interview, Mr. Tueller, 63, said he had simply wanted to warn officers that they might be in danger far sooner than they realized. Twenty-one feet as a justification for shooting, he said, just became a “sticky idea” in policing.

The Dallas police chief, David O. Brown, said at a policing conference in February: “Sometimes it seems like our young officers want to get into an athletic event with people they want to arrest. They have a ‘don’t retreat’ mentality. They feel like they’re warriors and they can’t back down when someone is running from them, no matter how minor the underlying crime is.”

Those remarks came just weeks before a police officer in North Charleston, S.C., was charged with murder for shooting an unarmed man in the back. The officer had stopped the man, Walter L. Scott, because of a broken brake light. When Mr. Scott ran, the officer gave chase, even though he had Mr. Scott’s driver’s license.

“In most cases, time is on our side,” Chief Whent, of Oakland, said in an interview. “We’re chasing someone whose name we know, and we know where they live.”