But here's the thing: I WANT to. And that doesn't seem to describe someone with schizoid personality disorder. I WANT to have friends

Well, you've given me enough to think I shouldn't discount the possibility that I'm schizoid. I wonder if that's why I sometimes feel like my schizophrenic roommate and I have a lot in common. I know they're not really related, but it's the same word, there must be some connection...

You could be an INTx in myers Briggs. INTPs in particular can be quirky in their own right. They like puzzles and just figuring out whatever interests them. Maybe look up some INTP profiles (there are a few and they'll give different insights).
INTPs aren't the most common personality, so we can definitely feel like the odd one out. We process information and interact with the world differently than most.