I always introduce myself as "Hi, I'm ____, one of the medical students working in the ER / with the nephrology team / etc." I think the term student doctor is confusing and tends to convey more authority than we actually have.
That said, I have plenty of patients that don't remember my title and call me Dr. Sanityonleave and I don't correct them. I do make it clear when they ask me to explain something to them that I can clarify things for them but that they'll need to talk to one of the supervising doctors to find out an opinion on why we're doing x instead of y, etc. I think some medical students are too shy about accidentally voicing their opinion that they shy away from ever talking about the treatment plan and etc with patients. Often what patients need is an explanation of what's going on, which you're qualified to do (as long as you actually know what's going on). If it comes to opinions on x vs y, that's when you need to defer to your resident/attending.
Anyway, back on topic: student doctor is dumb. I go with medical student and have never had a problem. Some attendings do introduce me as Dr. Soandso, though, which is awkward. That said, correcting people doesn't do any good -- in general, they have no idea about the system anyway.