I just use "BP" since a lot of people online are familiar with that acronym... Here, Reddit, other forums... So online at least, it's pretty standard to use BP. In real life, though, I actually say "bipolar."

I couldn't care less if someone wants to say "I am bipolar" vs "I have bipolar," though. I don't think either one is "wrong" per se because it seems like 50% of people use "am" and the other 50% use "have," and everyone has their reasons for their preference. Besides that, at the end of the day, the only thing the person is trying to do is say "I have been diagnosed with bipolar disorder." I do not get offended, confused, upset, or bothered with however someone wants to convey that message, as long as they have good intentions.