Mark K. Updegrove, a presidential historian who wrote a book about Mr. Bush and his father, President George Bush, said neither of those commanders in chief nor President Barack Obama would have been so loose with the truth about so momentous an occasion. Events like the killing of Osama bin Laden, he said, were treated with solemnity and restraint.

“In the days of reality television, humility is not enough,” said Mr. Updegrove, who is also president and chief executive of the Lyndon B. Johnson Foundation in Austin. Mr. Trump, the reality television veteran, “has to continue to add to the inherent drama of the moment, not only bragging about the despot being brought to justice but happening in the most humiliating way. He can’t help himself.”

By contrast, Mr. Obama gave few details in announcing the raid that killed bin Laden in 2011, leaving that to official briefers. Even then, he and his team were criticized for being imprecise at first. Mr. Obama said bin Laden was killed “after a firefight” and the White House later said Navy SEALs “were engaged in a firefight throughout the operation.” But in fact, the only shots fired at the Americans came at the beginning of the raid and bin Laden himself never opened fire. Officials attributed the imprecision to the fog of war as they gathered information from the field.

As for Mr. al-Baghdadi, it is certainly true that there was no mourning his death in Washington this week. During his five-year reign, the Islamic State proved itself one of the most brutal terrorist organizations on the planet, establishing a murderous rule over millions while beheading and raping captured Americans.

Nor did anyone argue with the characterization of Mr. al-Baghdadi as a coward after he detonated a suicide vest with children nearby on Sunday, killing them along with himself.

But that was apparently not enough for Mr. Trump. In his announcement and subsequent question and answer session, he used the word “whimpering” six times, “crying” five times and “screaming” four times.

Mr. Trump’s account, aspects of which he repeated Friday evening at a rally in Tupelo, Miss., has left four-star generals in the awkward position of not confirming assertions they do not know to be true while trying not to contradict the president too overtly.