The major difference now, Mr. D’Antonio said, is that as president Mr. Trump is fact-checked assiduously.

Mr. Trump’s latest tangle with the truth began on Monday, when he said at a cabinet meeting that Mr. Peña Nieto had been on the phone to him. “Even the president of Mexico called me,” Mr. Trump said, touting his success in cracking down on illegal immigration. “They said their southern border — very few people are coming because they know they’re not going to get through our border, which is the ultimate compliment.”

The Mexican government said on Wednesday that no such telephone call took place. In a statement, Mexico’s secretary of foreign relations said Mr. Peña Nieto told Mr. Trump during the Group of 20 summit meeting that deportations of Mexicans from the United States had fallen 31 percent over the first six months of the year, compared with the same period in 2016.

On Tuesday, it was the Boy Scouts’ turn: A leaked transcript of an interview the president had with The Wall Street Journal quoted him saying that the head of the Boy Scouts had called him full of praise for a highly political speech Mr. Trump had delivered at the National Scout Jamboree.

“I got a call from the head of the Boy Scouts saying it was the greatest speech that was ever made to them, and they were very thankful,” Mr. Trump told The Journal. On Wednesday, the Boy Scouts of America said it was not aware of any call from its leadership to Mr. Trump. In a statement, the organization said that an earlier statement from Michael Surbaugh, the organization’s chief, apologizing to scouts for Mr. Trump’s speech, “speaks for itself.” Mr. Surbaugh had expressed regret to those who were “offended by the political rhetoric that was inserted into the jamboree.’’

It is hardly unprecedented for a president to use a story to inspire or motivate, or to embellish a yarn for the sake of punctuating a poignant message. President Lyndon B. Johnson was a frequent and animated storyteller, and Ronald Reagan was so partial to a heart-tugging anecdote that his tales sometimes aroused suspicion that they had come from a movie in which he had starred rather than real life.

For his first inaugural address — the first to be delivered from the West side of the Capitol facing Arlington National Cemetery — Mr. Reagan wanted to recount the story of a World War I soldier, buried in Arlington, who had written in his journal about his pledge to give everything for his country and died in battle the next day. The only trouble, his speechwriter told him, was that the fallen soldier was buried in his hometown, not at Arlington, according to H. W. Brands, a historian at the University of Texas and biographer of Mr. Reagan.

But the president, enamored of the story, left it in his speech, and said the soldier was buried “under one such marker,” leaving his actual resting place vague. The White House later conceded that the man in question was not under a marker at Arlington.