“He thinks about destiny in human terms,” Mr. Wiesel said in an interview.

Still, others say the Obamas have become more relaxed schmoozers, more at ease with the porous line between the political and social, more willing to reveal themselves. They have recently begun inviting more outsiders into their private living quarters, including Mr. Kushner, Steven Spielberg and Daniel Day-Lewis at the “Lincoln” dinner. At a dinner in late November to thank top campaign fund-raisers, the first couple was like a bride and groom, bantering and traveling from table to table to accept congratulations and good wishes for the years ahead, making sly jokes that guests would not repeat for publication.

Even Mr. Obama’s speech has changed a bit, close observers say. Though he still disdains Washington, he often sounds less like a disapproving outsider and more like a participant. One former aide was startled to hear Mr. Obama use “impact” as a verb, a particular tendency in the capital. Another longtime adviser said he was struck during the 2011 debt ceiling negotiations when Mr. Obama grew offended that House Speaker John A. Boehner did not return his multiple phone calls. The old Barack Obama would have thought the who-calls-whom protocol was stupid, the adviser said, but “the world that he inhabits now is the world of inside-the-Beltway maneuvering.”

In video footage of Mr. Obama as president, the contrasts can be subtle but amusing. At his first Thanksgiving turkey pardoning ceremony in 2009, the president played along, but then paused to distance himself from what he was doing and hint that he found the tradition ridiculous. “There are certain days that remind me of why I ran for this office,” he said. “And then there are moments like this — where I pardon a turkey and send it to Disneyland.” Cut to the same ceremony, three years later, and cue Mr. Obama promoting a contest over which bird to pardon and giggling just a little. “You can’t maintain your day-to-day cynicism about that stuff,” said Arun Chaudhary, the former White House videographer, because many Americans cherish White House rituals.

What Mr. Obama wants to achieve this term is pretty clear: a fiscal deal and overhauls of gun and immigration laws, steps to address climate change and less restrictive voter identification laws. But what Mrs. Obama wants is more of a mystery. In almost every appearance, she sounds warm, unpretentious notes; on Friday, she continued her Twitter banter with Ellen DeGeneres over who could do more push-ups.