Oh, bisexuality — like most things— exist on a spectrum and my personal experience will be different from yours will be different from someone else‘s, of course. And I definitely don‘t ever want to gatekeep who is „allowed“ to call themselves bi, though I also don‘t think that‘s what I did.
I‘ve heard the terms „gender indifferent“ and „gender conscious“ used to describe the different kinds of bi-/pansexuality. It seems you are gender indifferent, I am gender conscious. Both are equally bi 
But yes, this is getting very theoretical and belongs in another thread.