Have you ever explored asexuality? There are different forms of it, it's worth a look since this is how you feel. I have several friends that consider themselves asexual who also consider themselves atheist or agnostic.
I had the same doubts about the term bisexual as you. I can't stand the gender binary...I'm queer to gender. I don't like saying omnisexual or pansexual because of the connotations of promiscuity, a stereotype which we already deal with enough already. I stumbled upon a blog post talking about genders and the term bisexuality that informed me about something which allowed me to accept the term bisexual.
Consider that homo=same and hetero=different.
Different-sex attraction, or heterosexual, doesn't strictly mean opposite gender attraction. So, bisexual is homosexual (attracted to one's own gender) and heterosexual (attracted to other genders).
This kind of blew my mind. I know it's not going to come across well on a poster either. ;-)
As for the, I was x years old stories, I know they're hard to believe if you don't have that experience, but I was closeted and in denial from a young age until college. In first grade I have a distinct memory of when family friends were visiting from across country that I had the epiphany that I liked women sexually. The rarity of their visits corroborates the time. I don't know how I knew the word lesbian, but I wrote "I am a lesbian" while hiding the words from my family and friends. It is a very detailed memory and an experience I'm sure very few heterosexuals have.