No, bisexuality doesn't cover anything well. That's why we have labels like pansexual, polysexual, and omnisexual now, which many of us find fit far better than bisexual.
Again, they're not labels, they're descriptive terms, as are your examples. I agree that bisexuality can be too vague a term to describe the experience of many people. Do you agree that you are polysexual because of who you are attracted to, not because you identify as polysexual?
