Protesters were thousands-thick in Tompkins Square Park in Manhattan’s East Village when the police moved in with horses and nightsticks. The tactics were described by a labor leader as “an orgy of brutality” and brought a public outcry demanding that police officials be fired.

This was not a Black Lives Matter protest in 2020, or even the riot that erupted in the same park in 1988 as officers charged at protesters. This head-knocking happened during a demonstration by unemployed workers amid the financial panic of 1873.

New York has long been one of the biggest stages for protest in the United States, with a vocal, sometimes volatile populace and a rich tradition of dissent. How the police handle such demonstrations in the densely packed city has been a source of debate since the department’s founding 178 years ago, and its actions have sometimes shaped American history.

On Tuesday, a proposal for a sweeping agreement was filed in federal court between police officials and the state attorney general that might strike a new equilibrium. Though the city’s largest police union opposes the deal and the judge overseeing the case has said she will hear its complaints, the plan has created tentative hope among civil libertarians.

“The N.Y.P.D. has historically been very, very bad about learning from its errors,” said Corey Stoughton, a lawyer for the Legal Aid Society, which was a plaintiff in the lawsuit that led to the agreement. “You see the same mistakes being made over and over. You see mass arrests. You see patterns of violence. Here, the hope is to break exactly those cycles.”

The agreement seeks to end a pattern of overwhelming police responses that can lead to violent clashes with protesters, injuries on both sides, large-scale arrests and, many times, civil rights lawsuits. The city paid $121 million last year to resolve suits alleging police misconduct, the most since 2018. The total was driven up by several large exoneration payouts and settlements stemming from the Black Lives Matter protests three years ago.

Mayor Eric Adams, a former police captain, said New Yorkers’ rights to both peaceful protest and public safety were “essential for our city to function” and that “balancing these two important rights is one of this administration’s core missions.”

The settlement announced this week would require the Police Department to rewrite policies, carry out training and use a tiered system of de-escalation techniques before deploying officers. A high-ranking department executive would be assigned to ensure compliance with the new rules.

An oversight committee responsible for reviewing how the changes are adopted and employed would include representatives from the attorney general’s office, the New York Civil Liberties Union and the Legal Aid Society, as well as the city investigation commissioner, city lawyers and a police executive assigned to supervise protest responses.

The proposal came after more than two years of litigation and at least three dozen mediation sessions, but it is not yet complete.

On Friday, Judge Colleen McMahon said in an order that after having initially approved the settlement Tuesday, she was revisiting that decision after the Police Benevolent Association, which had intervened in the case with other unions, said it had not had the chance to weigh in. She said that neither the union’s lawyer nor any other attorneys had told her of doubts about the deal, so she would let the union argue against it later this month.

“I should have been told that there was opposition to the case months ago,” Judge McMahon wrote, adding: “I am not going to put myself in a position to be reversed.”

After hearing objections from the union, the judge could approve the settlement as it stands, reject it or perhaps order further negotiations.

The P.B.A. president, Patrick Hendry, said in a statement that almost 400 officers had been injured in 2020 when looters stormed through Midtown Manhattan, SoHo and parts of the Bronx amid the Black Lives Matter protests.

“The individuals and groups responsible for the 2020 violence and destruction will surely view this agreement as a green light to create more of the same,” he said.

Brian Higgins, a former chief of the Bergen County, N.J., police department and an adjunct professor at John Jay College of Criminal Justice, said the settlement could make it harder for officers to respond quickly if a crowd suddenly became “very violent, very dangerous and very hard to control.”

The city expects to pay a total of roughly $50 million to settle suits related to the 2020 protests, according to estimates from the city comptroller’s office.

The sum includes $415,002 for two demonstrators — state lawmakers — who accused officers of penning them in and pepper spraying them. In another settlement, several volunteer legal observers received $349,012 after claiming officers violently arrested them.

And in July, the city agreed to pay about $13.7 million to settle a federal class-action suit filed by about 1,380 protesters who said they had been subjected to unlawful police tactics in 2020.

Switching to an approach that emphasizes dialogue over force will be “very difficult” and require considerable training, said Edward Maguire, a criminology professor at Arizona State University who was a consultant to city lawyers during their assessment of the 2020 protests. But, he said, it was “very important” that the settlement lead to real change.

“The N.Y.P.D. is the largest police agency in the United States,” Professor Maguire said. “It needs to reform how it does this and set a better example for the rest of the country.”

Micaela Martinez, 38, a plaintiff in the lawsuit that led to the settlement announced this week, was arrested amid a protest in the Mott Haven section of the Bronx in June 2020. She and other protesters were detained for hours after being surrounded by a throng of officers and herded into tight spaces on the street in a tactic known as kettling.

“They push in on you so tightly that you can’t breathe,” said Ms. Martinez, who recalled officers standing on the roofs of cars and beating protesters over the heads with batons.

“It’s going to be a lot easier in the future to make them accountable,” Ms. Martinez said. “It does feel like a win.”

James Lauren, a nurse practitioner who was at the Mott Haven protest as a volunteer medic and a plaintiff in the suit, said he was haunted by images of injured, bleeding protesters.

Mr. Lauren, 41, said he had been dressed in scrubs, carrying a clear case that showed his medical supplies and not participating directly in the protest. Still, he said, the police had handcuffed him and refused to let him tend to those who had been hurt.

“If they’ll even arrest the Legal Aid observers, nurses who are in scrubs, how is anybody supposed to feel like they’re safe?” Mr. Lauren said.

The Police Department has faced criticism for its responses to protests going back nearly to its founding in 1845.

“At least since the Civil War, there have been ongoing riots, demonstrations and protests in the city, and generally the police wind up thinking they’ve done a good job quelling them, and the public takes the opposite view,” said Bernard Whalen, a former New York City police lieutenant who has written books on the department’s history.

In 1863, protests in Lower Manhattan against the Civil War draft devolved into mob attacks on Black residents, driving many to Brooklyn and beyond. The police quelled days of deadly violence with the help of militia units.

More than a century later came the 1969 police raid at the Stonewall Inn in Greenwich Village that touched off a rebellion that transformed the gay liberation movement.

In 2011, the police staged a surprise raid after midnight to clear Zuccotti Park in Lower Manhattan of Occupy Wall Street protesters who had turned the park into their encampment and the center of the protest.

The police were criticized for pepper-spraying protesters and for ultimately storming the park in riot gear and armed with lights and loudspeakers. They arrested about 200 people and removed protesters, along with their tents, tarps and belongings.

By contrast, the police in Boston and Salt Lake City dispersed protesters who had set up encampments far more peacefully after leaders spent days developing relationships even as they warned protesters they would soon be ordered to leave.

In Boston, William Evans, a police superintendent who later became commissioner, visited so often that he was on a first-name basis with many of the protesters and later wrote some of them college and job recommendations, according to a 2018 report on mass demonstrations by the Police Executive Research Forum.

“The graduated response is something police understand,” said Chuck Wexler, the forum’s executive director.

Responses that begin with dialogue are the most effective way to keep officers and protesters alike safe, said Professor Maguire of Arizona State.

“Nobody is saying that these officers can’t protect themselves,” he said. “It’s just not triggering a fight or instigating a fight. When you show up in all sorts of force and in tactical gear, in many ways it becomes a self-fulfilling prophecy.”

“You go there looking for a fight,” he added, “you’re going to get a fight.”