My definition of geek: someone with a passion for a non-mainstream subject, hobby, interest or field of study, or someone with a non-mainstream level of interest in a mainstream subject, hobby, interest or field of study.
In my opinion, self identified geek girls are offended by fake geek girls because the fake geek is displaying a shallow, outdated and somewhat prejudiced understanding of what a geek actually is. The hallmarks seem to be that they have a pair of thick rimmed glasses, they may have read at least one book at some point in their lives, and they may refer to themselves as being socially awkward even though this may not really be the case. For the girl who considers herself a true geek, her eyewear, literacy and social life are probably not the criteria that she feels qualify her as a geek.
Then, to pile on further insult, the fake geek girl is quick to disassociate herself from those elements of geekdom that she thinks are beneath her. Just as the author in the linked article did, she trashes geek fiction and movies. The message the fake geek girl is sending to geek girls is “I’m just like you! I have ugly glasses and I read the Hunger Games once, and last weekend when I was broke I stayed home and Facebooked instead of going out clubbing… until my friends dragged me out. I’m also a gamer - you don’t want to know how many hours I’ve spent in The Sims! We’re exactly the same, except everything you like is stupid.”
I don’t think the hate is directed at the fake geek girl for mislabeling herself. It’s because it exposes what she really thinks a geek is, and she is trying to latch on to some perceived cachet of a group that she simultaneously shows contempt towards. Why does she want to be part of that culture? I think she’s often a girl who perceives herself as smart. Geeks are smart, so she sees herself as a geek but, like, a cool geek and not a pathetic Dr Who watching, anime enjoying, science fiction reading geek.
To top it off, the casual misogyny of some male geeks makes female geeks feel like they aren’t taken seriously within the subculture. The fake geek girls fulfill every negative stereotype the misogynist geeks have of female geeks, and their very existence further undermines the credibility of women who are actually serious about their passion for all things nerdy. On the one hand, geek girls get limited with all the prejudiced stereotypes of being awkward, socially impaired and nerdy from society at large, while on the other they are treated like they don’t belong and are somehow lesser by some of their male counterparts.