PUEBLO, Colo. — After obliging a request for a selfie from a woman in the lobby, Bill Clinton left his hotel here on Friday and rode to his rally, where he made the case that Hillary Clinton cared more about restoring jobs to forgotten workers than Donald J. Trump did.

Mr. Clinton said that his wife got things done, as evidenced by “a lady in a hotel lobby right here.”

“I said, ‘Who do you think we ought to elect?’” Mr. Clinton said, supposedly recounting his conversation with the woman. “She said, ‘You don’t have to ask me that. I’m from Belfast. I remember what she did to help make the peace in Northern Ireland.’”

It was Mr. Clinton, not his wife, who played a major role in brokering a peace accord. And the woman in the lobby, Lorraine Gordon, who immigrated from Belfast and is now an American citizen, never said those things.