In the May 8 Democratic primary for governor of Ohio, two well-known progressives, Richard Cordray and Dennis Kucinich, are running on differing brands of liberalism that reflect divergences on the left.

AKRON, Ohio — Richard Cordray speaks softly and carries a big stack: lime-green index cards, pressed into his shirt pocket, near enough for any sudden onset of note-taking.

A former director of the Consumer Financial Protection Bureau, he has been endorsed in his bid for Ohio governor by Senator Elizabeth Warren, who has told him he needs to learn how to brag more. “I am pretty good at getting back people’s money,” Mr. Cordray managed before an outdoor crowd of dozens here recently. Polite applause followed. He is trying.

Dennis Kucinich speaks until someone interrupts him — and even this is often insufficient — and carries a bag of vegan groceries heavy enough to sink his right arm like a weight-bearing scale of justice.

A former congressman and presidential candidate also running for governor, he has been endorsed in the May 8 Democratic primary by allies of Senator Bernie Sanders, who remember Mr. Kucinich’s lectern-pounding opposition to the Iraq war and his long-ago turn as the “boy mayor” of Cleveland.

“Dennis!” shouted Keith Thornhill, 58, a cape-wearing sidewalk busker who was a teenager when Mr. Kucinich, now 71, first ran the city. “Glad you’re back in the game, man.”

“Hey, Superman,” Mr. Kucinich said.

Dennis Kucinich pushed ideas like tuition-free college and single-payer health care long before they came into wider fashion.

Since the election of President Trump, certain conflicts have been inevitable for a Democratic Party asking itself how to win again: liberal or moderate candidates? Populist or pragmatist? Establishment or insurgent? But in the race between Mr. Cordray and Mr. Kucinich — one of the year’s most closely watched Democratic primaries — a more basic tension has consumed the collective left: Who has the truest claim to progressivism in 2018, when both candidates can credibly grab at the label? Is it better to be liberal on guns (Mr. Kucinich) or the bane of the banks (Mr. Cordray)? To be a fire-breather or a bit of a square?

“There’s no stigma in being competent,” said Kevin Davis, 63, of Akron, a Cordray supporter and fund-raiser who has gravitated toward the candidate’s work-within-the-system defense of responsive government. “Dennis just promises everything.”

“It’s hard not to be pulled into the vortex of Dennis Kucinich,” said Nina Turner, a former state senator and the president of Our Revolution, a group that was formed out of Mr. Sanders’s presidential campaign and that has endorsed Mr. Kucinich. “He’s infectious.”

As Democrats look to reclaim a purple state shading red, recent history suggests a slog. Mr. Trump won Ohio by eight points. It has a Republican-controlled legislature and has had a Republican governor for all but four years since 1991. But the prospect of a Democrat-friendly election year and a whiff of scandal in Columbus — where the House speaker resigned this month as federal investigators questioned his conduct — has convinced strategists that even a proudly liberal candidate could win in November.

The Democratic primary has also doubled as an early proxy test for supporters of Ms. Warren and Mr. Sanders, two possible presidential candidates in 2020, and a peek at precisely what kind of figure can speak to today’s party base in a Midwestern bellwether.

Mr. Cordray, 58, a former state attorney general before his time in Washington, has campaigned with Ms. Warren, who devised the bank-regulating agency he oversaw under President Barack Obama. He is leaning heavily on support from labor groups like the Ohio Federation of Teachers and the Ohio A.F.L.-C.I.O., specking public appearances with protect-the-little-guy anecdotes from his watchdog role in the hope that voters will not mistake boring for moderate.

There is talk of task forces and pension…