Editor's note: Peter Bergen is CNN's national security analyst, the author of "Manhunt: The Ten-Year Search for bin Laden -- From 9/11 to Abbottabad" and a director at the New America Foundation. Watch more at 4 p.m. ET on "The Lead with Jake Tapper"

(CNN) -- In February, Esquire magazine published a lengthy profile of "The Man Who Killed Osama bin Laden." The story did not identify the killer by his real name, referring to him only as "the Shooter."

The Shooter told Esquire that the night bin Laden was killed he had encountered al Qaeda's leader face-to-face in the top-floor bedroom of the compound in Abbottabad, Pakistan, where bin Laden had been hiding for more than five years.

The Shooter explained that when he found bin Laden in his bedroom the al Qaeda leader was standing up and had a gun "within reach" and it was only then that the Shooter fired the two shots into bin Laden's forehead that killed him. That account was in conflict with the account from another raid participant in a wildly successful book "No Easy Day."

Now, another member of the secretive SEAL Team 6, which executed the bin Laden raid, tells CNN the story of the Shooter as presented in Esquire is false. According to this serving SEAL Team 6 operator, the story is "complete B-S."

SEAL Team 6 operators are now in "serious lockdown" when it comes to "talking to anybody" about the bin Laden raid and say they have been frustrated to see what they consider to be the inaccurate story in Esquire receive considerable play without a response. Phil Bronstein, who wrote the 15,000-word piece about the Shooter for Esquire, was booked on CNN, Fox and many other TV networks after his story came out.

Twenty-three SEALs and their interpreter assaulted the bin Laden compound just after midnight on the morning of May 2, 2011. They shot and killed bin Laden's two bodyguards, one of bin Laden's sons and the wife of one of the bodyguards and they also wounded two other women.

The first three SEALs to make it to the top floor of bin Laden's compound where he was believed to be living were "the point man," "the Shooter" profiled by Esquire, and Matt Bissonette, the SEAL who wrote "No Easy Day" under the pseudonym Mark Owen.

What actually happened the night of the raid, according to the SEAL Team 6 operator who I interviewed, is that the "point man" ran up the stairs to the top floor and shot bin Laden in the head when he saw what looked like bin Laden poking his head out his bedroom door. The shot gravely wounded al Qaeda's leader.

Having taken down bin Laden, the point man proceeded to rush two women he found in bin Laden's bedroom, gathering them in his arms to absorb the explosion in case they were wearing suicide vests, something that was a real concern of those who planned the raid.

Two more SEALs then entered bin Laden's bedroom and, seeing that al Qaeda's leader was lying mortally wounded on the floor, finished him off with shots to the chest.

This account of bin Laden's demise is considerably less heroic than how the Shooter is presented in Esquire, in which he says he shot bin Laden while he was standing up and only after he saw that al Qaeda's leader had a gun within reach.

The SEAL Team 6 operator who spoke to me says there is no way the Shooter could have seen a gun in bin Laden's reach because the two guns that were found in bin Laden's bedroom after al Qaeda's leader was killed were only found after a thorough search of the room and were sitting on a high shelf above the frame of the door that opened to bin Laden's bedroom.

The SEAL operator also points out there was a discussion before the raid in which the assault team was told "don't shoot the guy [bin Laden] in the face unless you have to" because the CIA would need to analyze good pictures of bin Laden's face for its facial recognition experts to work effectively. Yet the Shooter in the Esquire story says he shot bin Laden on purpose twice in the forehead.

The SEAL Team 6 operator also tells CNN that the Shooter was "thrown off" of Red Squadron, the core of the SEAL Team 6 group that carried out the bin Laden raid, because he was bragging about his role in the raid in bars around Virginia Beach, Virginia, where SEAL Team 6 is based. In the Esquire article, Shooter complains that he is receiving no pension, since he left the military four years before the minimum twenty required to be eligible.

CNN spoke with Phil Bronstein, the Esquire writer, who says he passed on CNN's written questions about the Shooter's role in the bin Laden raid to his story's main character. The Shooter has not responded to those questions and Bronstein, himself, declined to be interviewed on-the-record for this story.

According to present and former members of SEAL Team 6, the "point man" who fired the shot that likely mortally wounded bin Laden will never "in a million years" speak publicly about his role in the raid. All laud the point man for his courageous decision to throw himself on the two women in bin Laden's room.

The new account of the night of the bin Laden raid provided by the serving SEAL Team 6 operator is essentially the same as in Bissonnette's "No Easy Day." Bissonnette says he was one of the first to run into bin Laden's bedroom and he saw that the point man's shots had mortally wounded bin Laden, and Bissonette then shot the dying al Qaeda leader as he lay on the floor.

Present and former members of SEAL Team 6 say they regard Bissonnette as more credible than the Shooter.

In a previous CNN.com story about the Esquire profile, I noted that I was the only outside observer allowed to tour bin Laden's Abbottabad compound before it was demolished in late February 2012.

During that tour I looked around the bedroom where bin Laden was killed. The Pakistani military officers who were guiding me pointed out a patch of dark, dried blood on the low ceiling of bin Laden's bedroom. This patch of congealed blood seems to be consistent with the Shooter's story that he fired two shots at the forehead of a "surprisingly tall terrorist" while he was standing up. At the time, the precise location of bin Laden when he was shot was not a matter of dispute.

But the blood patch could also be consistent with the account that it was the "point man" who first shot bin Laden. The point man is 5 feet 6 inches tall and was shooting upward at a tall man as he poked his head out of his bedroom.

The compound is, of course, now gone, so it is no longer possible to reconstruct what happened the night of the raid based on forensic evidence, although it is possible the Abbottabad Commission, a panel that was appointed by the Pakistani government to look into the raid, could shed some light on this question should its findings ever be publicly released.

Finally, by all accounts, it was a confusing situation the night of the bin Laden raid in Abbottabad. One of the SEAL team's helicopters had crashed and there was then a firefight with one of bin Laden bodyguards. All the electricity in the bin Laden compound and indeed the surrounding neighborhood was off on a moonless night and the SEALs were all wearing night vision goggles that allowed them only quite limited vision.

What seems incontrovertible is that the point man,the Shooter and Bissonnette were the first three SEALs to assault bin Laden's bedroom. But to determine exactly which of them killed bin Laden may never be possible.

What is certain is that it was a team effort.

Five days after the bin Laden raid, members of the SEAL team who killed al Qaeda's leader briefed President Obama. According to those in the room, the SEAL team commander explained to the president, "If you took one person out of the puzzle, we wouldn't have the competence to do the job we did; everybody's vital. It's not about the guy who pulled the trigger to kill bin Laden, it's about what we all did together."