They are both trailing behind Joseph R. Biden Jr., the former vice president, in the Democratic race; Mr. Biden has run on a far more conventional message of making gradual policy improvements from the center-left, and he has expressed reservations about the rhetoric of his more liberal rivals about corporations and billionaires. The party is currently locked in a grand debate over how best to build an electoral majority, and whether Democrats would be better off appealing to voters with a soothing promise of returning to normalcy or with a more activist message about economic and social injustice.

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These divergent strains of populism are far from new in American politics: for much of the country’s modern history, mass social movements channeling grievances with government or big business have competed with other forces directing outrage at racial and cultural minorities, immigrants and foreign countries.

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To some Democrats, the task of delivering a credible message of changing a broken system in Washington is a defining challenge of the 2020 election. Tiffany Muller, head of the influential clean-government group End Citizens United, said her organization’s research showed that many swing voters still see Mr. Trump as a political outsider with what Ms. Muller called an undeserved veneer of ethical independence.

“What we’ve seen is that Trump actually maintains strength on this issue — that, frankly, voters don’t know who to trust on the issue of corruption and cleaning up Washington,” Ms. Muller said in an interview on Monday afternoon. “We have got to go after his strength on this issue and win back some of the voters we lost in 2016.”

Ms. Warren proposed a battery of new reforms in her remarks in New York City’s Washington Square Park, near the 1911 Triangle Shirtwaist factory fire that she cited as an example of the oppression of the working class. And she highlighted an array of other reforms she has previously outlined, including a ban on lobbying by foreign governments and new ethics regulations on presidents and judges. She presented herself not just as an opponent of Mr. Trump, whom she called “corruption in the flesh,” but of the Washington system writ large.

“Too many politicians in both parties have convinced themselves that playing the money-for-influence game is the only way to get things done,” Ms. Warren said, vowing to do things differently: “No more business as usual. Let’s attack the corruption head-on.”

Mr. Trump’s version of populism is starkly different and, to most voters, already well known. While he has periodically taken rhetorical aim at certain big corporations, like pharmaceutical companies, he has largely abandoned early efforts to make good on his drain-the-swamp rhetoric from the 2016 campaign, and he has faced a barrage of ethical questions about the intermingling of his hotel and real estate business with the work of the government. He has invited business executives and lobbyists into his administration and a number of cabinet departments and agencies have drawn close scrutiny for potential conflicts of interest, including the Environmental Protection Agency and the Department of the Interior.