When people say don't let it define you, they mean don't let it push Ito controlling who you are or want to be. I'm 27 and I think I'm just starting to figure out who I am. You are a person who happens to have some chemical problems in your brain. for example, a lady had breast cancer, and it controls a lot other life, but if someone said define yourself in one word, she probably wouldn't say "CANCER!" You're young (I was 15 when I was diagnosed) but that doesn't mean you can't develop an identity outside of this disability. Think about kids who can't walk. There are lots of things they can't do, but they don't let that dictate who they are. Youre probably super creative, and because of your mood swings you can wear all sorts of hats. Use it to your advantage.
You seem to have a good insight into your condition, you are the one to decide about what/how you're affected. Some would be unwilling to have their bi-polar removed, as it can have some effects that they want. My experiences are mixed, but it does enable me to sometimes achieve goals that I'd not attain without it. The Downside's that it can be expensive in cash & emotional terms. Make plans, otherwise you'll just drift. You're OK.
I too have rapid cycling and I understand how you feel. That being said, you are you, you just happen to have an illness. Your personality is ingrained in the core of who you are. Don't let the bipolar define who you are no matter how moody or the diff phaseses of rapid cyclinging. Your young and you will contine to change as a person and develope. Even as adults we change and develope but our core person is always there.
Although I wasn't diagnosed till recently, I had symptoms of bipolar since I was 16. I'm BDII (not rapid cycling). It's hard, and for a while I did feel like what you described. I felt like I couldn't tell if I was happy and wanted to be adventurous or if I was just becoming hypomanic. I figured being 'depressed' was part of who I was, in terms of who I ended up hanging out with, the music I listened to, and the choices I made. But I can tell you that it did change for me. And I've found that I do have things that I really identify with, regardless of mood. Maybe you can try journaling to see any pattern in your choices. This way it can help you see if you like/ do something only when your manic/hypomanic/ depressed. You are still young and have a lot of time to figure out who you are =)... I think I'm still figuring myself out haha ...