Such shows of defiance and forcefulness by Mrs. Fiorina have impressed many liberal women.

Tracy Clark-Flory, a senior staff writer for Vocativ.com who has written about sex and relationships for Salon and other outlets, said Mrs. Fiorina’s debate turn was a powerful moment that created some dissonance given her stands on the issues.

“I think even as a lot of feminists cheered her on during that performance, we were loathing her actual policies,” she said. “There’s an excitement and a horror that those two can kind of coexist.”

Left-leaning feminists have not been so conflicted by other Republican candidates for national office in recent years. Both Sarah Palin, the Republican vice-presidential nominee in 2008, and Michele Bachmann, a congresswoman who ran for the Republican nomination in 2012, went out of their way to attack traditional feminism.

Mrs. Fiorina has spoken critically of traditional feminism, too: In June, when she still barely registered in national presidential polls, she declared in a speech that the liberal “version of feminism isn’t working” and said her definition of “a feminist is a woman who lives the life she chooses.”

That differs from the definition that Mrs. Clinton gave in an interview with the actress and writer Lena Dunham, the creator and star of the HBO series “Girls,” for a website Ms. Dunham is starting. When Ms. Dunham asked Mrs. Clinton if she considered herself a feminist, she replied, “Yes, absolutely,” and expressed puzzlement at women who did not. “A feminist is by definition someone who believes in equal rights,” she said.

Still, Mrs. Fiorina’s background as a former chief executive at Hewlett-Packard and her role in calling out Mr. Trump, whose remarks have offended women in both parties, have captured the attention of many writers, activists and other influential figures in the feminist movement, as reflected on social media and in news outlets targeted at young women. It was an irony first pointed out by the conservative blogger Michelle Malkin.

“Carly Fiorina is an ice-cold shade queen debate princess and I’m in love with and terrified of her,” Erin Gloria Ryan, managing editor of the feminist blog Jezebel, wrote on Twitter on Sept. 16.