President Donald J. Trump had just delivered his fiery speech at the Ellipse early on the afternoon of Jan. 6, 2021, setting in motion the attack by his supporters on the Capitol.

When he got into his armored vehicle after the speech, Mr. Trump immediately brought up a topic he often broached after his public appearances: How big was the crowd?

But within 30 seconds, his conversation with his lead Secret Service agent took a more contentious turn, according to a transcript released on Monday of an interview by House investigators of another Secret Service agent who was driving the car. Mr. Trump wanted to go to the Capitol, but his lead agent, Robert Engel, said no, telling him there was no plan in place.

“The president was insistent on going to the Capitol,” recounted the driver, whose name was not disclosed. “It was clear to me he wanted to go to the Capitol. He was not screaming at Mr. Engel. He was not screaming at me. Certainly his voice was raised, but it did not seem to me that he was irate — certainly not, certainly didn’t seem as irritated or agitated as he had on the way to the Ellipse.”

But, the driver said, Mr. Trump never lunged for the steering wheel or physically accosted the agents, contradicting one of the most sensational and hotly disputed elements of testimony given to the House Jan. 6 committee by a White House aide. The transcript of the driver is the first extensive eyewitness account of what happened in the armored vehicle to be made public.

“I did not see him reach,” the Secret Service driver told investigators for the House panel. “He never grabbed the steering wheel. I didn’t see him, you know, lunge to try to get into the front seat at all. You know, what stood out was the irritation in his voice, more than his physical presence.”