The push and pull over which path to pursue is playing out in tangible ways. In Iowa, interest groups are trying to push the Democratic candidates to the left. Greenpeace, for instance, stationed a field organizer in Des Moines who totes a hand-operated scoreboard to political events displaying the grades the group has awarded each of the 2020 candidates for their stances on the environment. In New Hampshire, volunteers with the American Civil Liberties Union appear at town hall events to press candidates to end the cash bail system and cut incarceration rates in half.

There are no organizations rallying early-state voters and candidates to the political middle.

The debate over what kind of candidate to run against Mr. Trump is a Rorschach test for how Democratic candidates, activists and voters see the future — and the past.

[It was a busy week in the 2020 race. Catch up on what you missed.]

The party’s center-left candidates argue that Mr. Trump is a historical comma, a four- or eight-year break from the country’s political baseline. They promise a return to a bygone political era of bipartisan cooperation and respectful political debate, with far less polarization.

“Trump is very much a symptom of our problems, not the cause,” Senator Michael Bennet of Colorado said in an interview earlier this month. “People thought, ‘We couldn’t do any worse, we might as well blow the place up.’ We need a better standard than that.”

Others, including Mr. Buttigieg, Ms. Warren and Mr. Sanders, view the president as the period at the end of an era in American political life.

“The Reagan era has basically defined my entire life span, and it’s finally ending,” Mr. Buttigieg said in a recent interview. “We’re just in a different place than the kind of ’90s formula where you could assume that part of how you appeal to independent voters was to pursue ideological centrism.”