Lily Potter is NOT a Muggle.

So I'm pathetically obsessed with Harry Potter, if you didn't already know.
I'm so bored today, and when I'm bored, I read the weirdest things online.
I love to read parodies, and one of my favorite ones is Movies in Fifteen Minutes. I was reading a new one that I found on Google, because I've read the MIFM about a million times-- and still fall out of my chair every single time I read it.
I found some new spoofs on the first three Harry Potter movies, and they're pretty funny.

Anyway, I was reading through, and it reminded me of something.
I was wondering if anyone could explain this to me:
Why does everyone say that Harry Potter is a half-blood, instead of a pure blood? Why do people say that Lily Potter was a Muggle? Just because she was Muggle-born, doesn't mean she was a Muggle herself.
A couple of times throughout the books, it is said that Harry is a half-blood. But both of his parents went to -- and met at -- Hogwarts. So apparently neither of them were Muggles, therefore Harry is a pure blood.

I know, I'm just being a nerd.
But can someone PLEASE explain this to me?

-Lindsay