“I know that many want this campaign to be about my past,” he said. “I get it. That’s the game. But this isn’t a game. Every one of you know, no matter who you’re for, know in your bones, this election is different.”

Some in the friendly audience made clear they were not interested in relitigating Mr. Biden’s record on race, even though the issue remains of significant importance to many in the party, including to some younger people of color.

“I thought her comments were unnecessary and out of place,” said Irvin Williams, 81, of Ms. Harris’s comments about Mr. Biden’s record on busing. He and his wife, who are African-American, said they were deciding between Mr. Biden and Senator Elizabeth Warren of Massachusetts. “If he said it, it’s how many years ago? If that was his position, then so be it. He’s done a lot of good things for people in this country.”

Eloise Conyers, who turns 64 on Sunday, said she was leaning toward Mr. Biden “for now,” but also likes Ms. Harris.

“I support her point of view,” Ms. Conyers said, adding that Mr. Biden did not “address it effectively” in the debate. But, she said, “I know from his record and what he has done, he really supports equal rights for others.”

[We tracked down the 2020 Democrats and asked them the same set of questions. Watch them answer.]

Mr. Biden also made clear that in contrast to some of his more liberal opponents, who support eliminating private health insurance in favor of more expansive government-supported health care coverage, he is not calling for revolutionary change on issues like the Affordable Care Act, noting his support for a public option. It was an implicit contrast with those who would support bolder change as part of proposals like “Medicare for all,” and a sign of his increased willingness to engage with his opponents rather than keeping his focus on President Trump.

“We don’t have time,” he said of those who want to “start over.”

At least one rival campaign is already signaling that it does not plan to let Mr. Biden move on from his past so easily.

“Every candidate’s record will (and should) be scrutinized in this race,” tweeted Ian Sams, the national press secretary for Ms. Harris. “It’s a competition to become President of the United States. There are no free passes.”