Or maybe you need to be forgiven for assuming that the girl doesn't understand what the term means among her peer group?

Feminism won. Young people overwhelmingly support--as you point out--the basic tenet of equality. What, then, is gained by labeling oneself a "feminist"? Why would you do that? That's roughly as useful as calling oneself an "abolitionist". Essentially everyone agrees now that slavery was a horribly evil practice. Your word needs to mean something else.

So what does "feminism" mean in contrast to the norm, when the equality ideal is practically universally accepted? Well...some more extreme variant, obviously.

So without seeing the video, I'm none too sure that the girl doesn't have it exactly right for her social context--she prefers the moderate view, which is traditional feminism, and the people who go out of their way to call themselves "feminists" in contrast are the ones who aren't about "humankind and a love for everyone".