The concept of Medicare for all has become popular with Democrats: 81 percent support it, according to a recent Kaiser poll. Yet voter opposition to surrendering the insurance they are used to led to a backlash over President Barack Obama’s repeated promise that “if you like your plan, you can keep your plan” after it proved false for several million people under his health law. Many Democrats are keenly aware of that backlash, and the 2020 presidential race will be the first where many of the party’s leading candidates will have to explain and defend the meaning of Medicare for all.

For now, as Ms. Warren demonstrated, many candidates do not want to wrestle publicly with the details. After Ms. Harris’s comment, her aides hastened to add that she would also support less sweeping changes to health care; like most other candidates, Ms. Harris declined an interview request. And by Friday, Mr. Booker, hours after announcing his presidential bid, sought to curtail the matter by offering a brisk “no” when asked if he supported eliminating private coverage.

Yet there is one likely 2020 contender who is thrilled to discuss Medicare for all.

Mr. Sanders, in an interview, did not mince words: The only role for private insurance in the system he envisioned would be “cosmetic surgery, you want to get your nose fixed.”

“Every candidate will make his or her own decisions,” Mr. Sanders said, but “if I look at polling and 70 percent of the people support Medicare for All, if a very significant percentage of people think the rich, the very rich, should start paying their fair share of taxes, I think I’d be pretty dumb not to develop policies that capture what the American people want.”

But Michael R. Bloomberg, the former New York City mayor who is considering a 2020 bid on a centrist Democratic platform, said it would be folly to even consider a single-payer system. “To replace the entire private system where companies provide health care for their employees would bankrupt us for a very long time,” Mr. Bloomberg told reporters in New Hampshire on Tuesday.

The Congressional Budget Office has not scored Mr. Sanders’s Medicare for All bill, but a study last year by the Mercatus Center of George Mason University predicted it would increase federal spending by at least $32.6 trillion over the first decade. The cost could be even greater, the study says, if the bill overestimated the projected savings on administrative and drug costs, as well as payments to health care providers.

The divide between Mr. Sanders, a democratic socialist, and Mr. Bloomberg, a Republican-turned-independent-turned-Democrat, reflects the large chasm in a party that has been reshaped by President Trump.

The president’s hard-line nationalism has simultaneously nudged Democrats to the left, emboldening them to pursue unambiguously liberal policies, and drawn independents and moderate Republicans to the party because they cannot abide his incendiary conduct and demagogy on race. These dueling forces have created a growing but ungainly coalition that shares contempt for Mr. Trump but is less unified on policy matters like health care.

And these divisions extend to what is wisest politically.

Liberals argue that the only way to drive up turnout among unlikely voters or win back some of the voters uneasy with Hillary Clinton’s ties to corporate interests is to pursue a bold agenda and elevate issues like Medicare for all.

“Those who run on incremental changes are not the ones who are going to get people excited and get people to turn out,” said Representative Pramila Jayapal of Washington, the co-chair of the Congressional Progressive Caucus.

And by preserving their options, Democrats risk alienating liberal primary voters, some of whom consider support for Medicare for all a litmus test.

“The center is not a good place to be on these policies anymore,” said Mary O’Connor, 61, a substitute teacher and horse farmer in Middleburg, Va., who wants a single-payer system. “I’ll be watching extremely closely, and I will most likely jump on board and volunteer for whoever it is that’s going to be the most forceful for this.”

But moderates believe that most Democratic primary voters are more fixated on defeating Mr. Trump than applying litmus tests — and that terminating employer-sponsored insurance would only frighten the sort of general election voters who are eager to cast out Mr. Trump but do not want to wholly remake the country’s health care system.

“Most of the freshmen who helped take back the House got elected on: ‘We’re going to protect your health insurance even if you have a pre-existing condition,’ not ‘We’re going to take this whole system and throw it out the window,’” said Kenneth Baer, a Democratic strategist.

While polling does show that Medicare for all — a buzz phrase that has lately been applied to everything from single-payer health care to programs that would allow some or all Americans to buy into Medicare or Medicaid — has broad public support, attitudes swing significantly depending on not just the details, but respondents’ age and income.

On the House side, a bill similar in scope to Mr. Sanders’s is under revision and will soon be reintroduced with Ms. Jayapal as the main sponsor. Other Democrats have introduced less expansive “Medicare buy-in” bills, which would preserve the current system but would give certain Americans under 65 the option of paying for Medicare or a new “public option” plan. Another bill would give every state the option of letting residents buy into Medicaid, the government health program for poor Americans.

The buy-in programs would generally cover between 60 and 80 percent of people’s medical costs and would require much less federal spending because enrollees would still pay premiums and not everyone would be eligible. Some proponents, like Senator Jeff Merkley, Democrat of Oregon, have described them as a steppingstone on the way to a full single-payer system; some of the Democrats running for president are co-sponsoring these “Medicare for more” bills as well as Mr. Sanders’s.

Mr. Sanders has suggested options to raise the money needed for his plan, such as a new 7.5 percent payroll tax and a wealth tax on the top 0.1 percent of earners. He has also predicted several trillion dollars in savings over 10 years from eliminating the tax exclusion that employers get on what they pay toward their workers’ insurance premiums, and other tax breaks.

But Robert Blendon, a health policy professor at Harvard who studies public opinion, said it would be wise not to delve into financing details for now.

“The reason it failed in Vermont and Colorado was taxes,” Professor Blendon said, referring to recent efforts to move to a near-universal health care system in those states, which flopped resoundingly because they would have required major tax increases. “But Democratic primary voters will not go deep into asking how these plans will work. What they will say is, ‘Show me you have a principle that health care is a human right.’”

The general election will be a different story, Professor Blendon added. If Ms. Harris were to become the Democratic nominee and keep embracing the idea of ending private coverage, he argued, “she’s going to have terrible problems.”

The difficulty for Democrats, added Ezekiel Emanuel, a former Obama health care adviser, is that many voters look at the health care system the same way they view politics. “They say Congress is terrible but I like my congressman,” as Mr. Emanuel put it.

According to the Gallup poll, 70 percent of Americans with private insurance rate their coverage as “excellent” or “good;” 85 percent say the same about the medical care they receive. The Kaiser poll found that the percentage of Americans who support a national health plan drops by 19 percentage points when people hear that it would eliminate insurance companies or that it would require Americans to pay more in taxes.

Among those who make over $90,000 a year — the sort of voters in the House districts that several Democrats captured in the midterms — those surveyed in the Kaiser poll were particularly wary of an all-government system: 64 percent in this income group said they would oppose a Medicare for all plan that terminated private insurance.

“My constituents are tired of bumper sticker debates about complex issues,” said Representative Lizzie Pannill Fletcher of Texas, a freshman from an affluent Houston district. “We don’t want ideologues in charge.”

In Vermont, where former Gov. Peter Shumlin shelved his ambitious plan for a single-payer system in 2014 after conceding it would require “enormous” new taxes, advocates for universal health care are now resigned to a more incremental approach.

Dr. Deb Richter, a primary care doctor who helped lead the state’s single-payer movement, said that while the Democratic field is “going to have to face the T word,” being upfront about the required tax increases, she now thinks phasing in a government-run system is a better approach.

“There’s ways of doing this that don’t have to happen all at once,” she said, pointing to a push in Vermont to start with universal government coverage for primary care only. “But you need to talk about the end goal: We are aiming for Medicare for all, and this is a way of getting it done.”