WASHINGTON — When President Trump told donors at a fund-raiser this past week that he had invented a fact during a conversation with Canada’s prime minister, the surprise was not that America’s leader makes things up, but that he openly admitted it.

Or maybe admitted is the wrong word. He actually seemed to boast about it.

In the furor that followed the disclosure of his remarks, attention focused on the impact on relations with Canada and whether the president was right or wrong in his assertion about trade. But the episode goes to the heart of a more fundamental debate about Mr. Trump: When does he know the things he says are false, and when is he simply misinformed?

Mr. Trump, after all, has made so many claims that stretch the bounds of accuracy that full-time fact-checkers struggle to keep up. Most Americans long ago concluded that he is dishonest, according to polls. While most presidents lie at times, Mr. Trump’s speeches and Twitter posts are embedded with so many false, distorted, misleading or unsubstantiated claims that he has tested even the normally low standards of American politics.

“His statement this week was another reminder of how cavalier he is with the truth,” said Bill Adair, the founder of PolitiFact, the Pulitzer Prize-winning, nonpartisan fact-checking website owned by the Poynter Institute. “He seems so willing to say whatever suits him at that moment regardless of whether it’s true. In all the time that I was editor of PolitiFact and in the time since when I’ve worked with fact-checkers all over the world, I’ve just never seen any political figure distort the truth so recklessly.”