I place the result of 'What's an egalitarian?' as better than the die roll as to whether they will take it as one or the other, or even a third interpretation. If I say egalitarian, and they can't define it, they'll probably go into a questioning mode, which helps make them more open to discussion. The number of people who could define it AND be against it would be minimal. On the other hand, feminist has become almost an epithet to a significant chunk of the English-speaking world. It's a word that shuts down empathy and critical thinking for those people, even if they couldn't define it, which your anecdote helps show it can be hard for people to do at any rate.

Don't get too caught up in nuance and small strains of academic philosophy here. The difference is clear:

Egalitarianism is a philosophy that asserts equality among all people. An egalitarian holds the belief or principle that all people are equal and should be treated equally.

Feminism is a social movement born out of the pervasive and systematic disenfranchisement, oppression and abuse of women, which holds to an egalitarian philosophy of equal rights between men and women. A feminist is an advocate for the equal rights of women.

One is an abstract idea that influences modern humanism, liberalism and democracy. The other is a struggle to make that ideal a reality, that has a different face in every different time and place that it's happening. Both are virtuous.