They were nonetheless acutely aware of how explosive the word could be. When Mr. Obama ran for president in 2008, Republicans tried to wound him by accusing him of waging “class warfare” to achieve wealth redistribution. That fall, the Republican presidential nominee, Senator John McCain, derided Mr. Obama as the “redistributor in chief” as he seized on Mr. Obama’s comments to an Ohio man later known as “Joe the Plumber” that he wanted to “spread the wealth around.”

Mr. Obama survived that episode and other instances when Republicans deployed old recordings of him using the word “redistribution” as evidence that he was a closet socialist. But Mr. Obama had learned a lesson.

After he took office, he cast his goal of rolling back President George W. Bush’s tax cuts for higher earners not as economic redistribution, but as the restoration of top-end rates from the Clinton years.

The Affordable Care Act was a similar semantic sidestep. The law targeted high earners, too, by raising their Medicare taxes enough to reduce their after-tax incomes by nearly 2 percent, according to the nonpartisan Tax Policy Center. That revenue helped finance coverage for those currently without insurance, who tend to have lower incomes and who in many cases will receive government subsidies to make their premiums cheaper.

And yet for those nervous about potential changes, the president promised stability. “If you like your current insurance, you will keep your current insurance,” Mr. Obama said the day he signed the legislation in March 2010, a promise he made repeatedly as the Oct. 1 opening day of the online health insurance marketplaces approached.

Hiding in plain sight behind that pledge — visible to health policy experts but not the general public — was the redistribution required to extend health coverage to those who had been either locked out or priced out of the market.

Now some of that redistribution has come clearly into view.

The law, for example, banned rate discrimination against women, which insurance companies called “gender rating” to account for their higher health costs. But that raised the relative burden borne by men. The law also limited how much more insurers can charge older Americans, who use more health care over all. But that raised the relative burden on younger people.