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Gov. John Kasich of Ohio, who spoke at a brunch during the Mackinac Republican Leadership Conference, greeted supporters on Saturday on Mackinac Island, Mich.Credit Carlos Osorio/Associated Press

MACKINAC ISLAND, Mich. — Emboldened by a heralded debate performance, Carly Fiorina brought an air of political star power to this Michigan resort island in Lake Huron on Saturday, captivating a gathering of party activists with her toughness and vision for Republican leadership.

Politicians such as former Gov. Jeb Bush of Florida and Gov. John Kasich of Ohio are well known in the state, but it was Mrs. Fiorina’s arrival that was met with the greatest anticipation at a summit meeting where Republican presidential candidates sought to escape the shadow that Donald J. Trump has cast over the campaign season and broaden the political map.

“We must win in 2016,” Mrs. Fiorina said in a keynote address to the Mackinac Republican Leadership Conference. “I am a fearless fighter. I will not falter, I have been tested and I will fight this fight.”

While Mrs. Fiorina has generated a burst of excitement since last week’s debate, some activists and donors expressed concern that Mr. Trump, the billionaire developer, was tarnishing the party’s brand. Others expressed hope that he might even quit the race before the Iowa caucuses.

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At the conference, Senator Ted Cruz of Texas, said that he wanted to get beyond political “food fights.”Credit Carlos Osorio/Associated Press

The candidates seeking the nomination aimed to capture the mantle of political outsider that American voters seem to be craving.

“If you see a candidate that Washington embraces, run and hide,” Senator Ted Cruz of Texas said in the soaring ballroom of the island’s Grand Hotel.

Some view Mr. Cruz, a favorite of social conservatives, as being too right-wing for a “purple” Midwestern state that has not backed a Republican in a presidential contest since 1988. But he pledged his commitment to competing in the region and drew a standing ovation after delivering a long list of actions that he would take as president to roll back the Obama administration’s policies.

Mr. Cruz’s oratorical polish contrasted with the folksy charms of Mr. Kasich, who has been trying to demonstrate that his Midwestern sensibilities can have appeal across state lines. As part of his goal to introduce himself to the public, Mr. Kasich shared stories of how he got started in politics and ordered heaping plates of crispy bacon for guests at a brunch at which he spoke.

Continue reading the main story Interactive Graphic Who’s Winning the Presidential Campaign? History suggests that each party’s eventual nominee will emerge from 2015 in one of the top two or three positions, as measured by endorsements, fund-raising and polling.

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“He seems like us,” said Kathy Crawford, a state representative. “Hard working and all that.”

Unlike some cattle calls where candidates woo voters by chowing down on pork chops and corn dogs, the Mackinac conference brings them to a place known to some as “Detroit’s Davos” — a carless, fudge-making mecca where people travel in horse-drawn carriages and gather to discuss policy.

There was chatter among those who gathered about the whereabouts of Gov. Scott Walker of Wisconsin, who had canceled his trip once to focus on Iowa before rescheduling and agreeing to come. Then bad weather struck and scuttled his plans at the last minute, leaving some to wonder whether he had lost interest in the state.

But it was Mrs. Fiorina who set off the most excitement.

Throngs of supporters met her when she arrived on the island, and she was cheered during a 30-minute speech in which she laid out her plans for more efficient government and defended her criticism of Planned Parenthood. The former Hewlett-Packard chief executive even took a subtle dig at Mr. Trump, quipping that leadership was not determined by “how big your office is, your airplane, your helicopter, your ego.”

Mary Kay Thayer, a trustee of Monroe County Community College, said of Mrs. Fiorina: “She’s strong, and she doesn’t take any guff.”

Although candidates are trying to move beyond the focus on Mr. Trump, his failure to correct a questioner who said President Obama is a Muslim presented them with another litmus test that they could not escape. Mr. Bush addressed the issue on Friday night, affirming that Mr. Obama is an American and a Christian and refusing to give credibility to those who say otherwise.

But Mr. Cruz continued his refusal to criticize Mr. Trump, saying he wanted to get beyond political “food fights” and focus on the needs of the American people. He even offered a note of thanks for Mr. Trump’s presence in the race, explaining that it has brought attention to his fellow Republicans.

Senator Rand Paul of Kentucky, who has been one of the fiercest critics of Mr. Trump, said that he planned to begin redirecting his criticism from Mr. Trump back to Hillary Rodham Clinton.

“I think he’s yesterday’s news,” Mr. Paul told reporters at a Lego exhibit that he was using as his local headquarters. “I think we’ve seen the crest of Donald Trump.”

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