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I have difficulty with the whole labeling concept myself. Two terms that might suit better than bisexual could be pansexual or demisexual?
Pansexuality suggests more that you'd be attracted to all gender identities, including trans and those that choose to not identify as either gender, therefore your attraction has little to do with a person's gender.
Demisexual suggests that you "don't judge a book by it's cover" and don't have any sexual attraction to the things that you can instantly assess, such as looks and gender etc. So once a demisexual gets to know a person, they become sexually attracted to the person through the relationship they've built with them.
Personally, I just don't like to get caught up in labels. I've noticed through friends and stories, attractions and orientations can change over time, my partner always considered herself straight until quite recently, it's easy to get caught up in calling yourself something and then feeling trapped. People change, and that's okay!
For the sake of simplicity in conversation, when I'm telling people (although similarly, I probably have about... 1 leg out of the closet) I usually just tell them about my partner, just saying like "my partner and I blah blah blah and she this and that" and just let them decide what I am - I think most just decide I'm a lesbian. If somebody asks my sexual orientation I usually just say bisexual, as it's just well understood.