The rising tension over Mr. Obama’s legacy, most notably the Affordable Care Act, is a new crack in the party’s orthodoxy, as an ascendant progressive wing that favors transformative change raises questions about the incremental policies of the former president. After a few candidates used the Detroit debate to demand that Mr. Biden account for Mr. Obama’s record on issues such as deportations and free trade, Mr. Biden was joined by some of the former president’s advisers, who chastised the critics for committing political malpractice.

“The more time we spend attacking President Obama and his record and the less time we spend on what Donald Trump is doing to this country only serves to help one person — Donald Trump,” said Stephanie Cutter, a former top Obama campaign aide. “Everybody on that stage needs to keep that in mind.”

David Axelrod, the architect of Mr. Obama’s political ascent, said Mr. Biden would be wise to “lean in” on his ties to the former president and to “confront those who are attacking the Obama policies.” But Mr. Axelrod added that Mr. Biden and the other center-left candidates in the race have so far proved unable to make a compelling, affirmative case for what he called “a reality-based candidacy” that was at once uplifting and pragmatic.

“We ran on ‘yes we can,’ and you’re not going to win on ‘no you can’t,’” he said, recalling the Obama catchphrase.

Ms. Feldman, the pollster, said that a number of Democratic candidates had begun to stand out from the field — including Senators Elizabeth Warren of Massachusetts, Kamala Harris of California and Cory Booker of New Jersey — but that the overall trajectory of the race remained as fluid as ever.

“I think it may be a little less clear today than it was a week ago,” she said.

Among the liberal candidates, Ms. Warren demonstrated her formidable intellect and hard-charging populism at the debates, and Senator Bernie Sanders of Vermont made a full-throated claim to being the architect and chief promoter of ambitious liberal ideas. They delighted many progressives but also left the moderate wing of the party uneasy with their embrace of ideas like eliminating private health insurance, part of an uncompromising agenda the centrists fear will lead to electoral ruin.