Mr. Trump, a man of endless contradictions, did not explain why he decided to tweet the video to his 56 million followers as he unveiled the farm bill or to have the audio played as he entered the White House’s South Court Auditorium to sign the bill. When asked who was in charge of the tweet, a senior administration official claimed ignorance — that person said White House aides were trying to focus on the farm bill’s contents, presumably before moving on to the next thing.

Mr. Trump, who only a day earlier had tweeted a message declaring victory over the Islamic State and announcing the pullback of troops in Syria — a decision that would prove to have a grave effect on his relationship with Mr. Mattis — took the podium and, with the world on his shoulders, provided the background on the theme song performance.

“That was from the Emmys,” Mr. Trump said proudly as members of his administration greeted him with a round of applause. “I sang ‘Green Acres’ and received a very nice award that night.”

Actually, he didn’t win an award.

That evening, Mr. Trump won “Emmy Idol,” a spoof competition based on “American Idol,” for his performance with Ms. Mullally. In a 2016 appearance on “The Late Show,” she recalled the experience as a “dumb thing that they did on the Emmys” that allowed viewers to call in to vote for their favorite segment. She said Mr. Trump called her with his congratulations the next day.

“If he felt that way about ‘Emmy Idol,’” she said, “how do you think he felt about the presidential election?”