I've never opened a patient encounter as "medical student." I learned very quickly not to. Occasionally a patient will glance at my ID and read it themselves, and from there it's all downhill. While technically the most accurate, "medical student" is the most confusing thing that you can possibly tell a patient. Usually I just say I'm with the surgical team. Not sure what I'll be saying on my future rotations, "I'm with the family practice team" doesn't have the same kick to it.
In an ideal world, every patient is entitled to know. But in this same ideal world, every patient would be required to understand the vast and complicated medical hierarchies - that the med student and the pgy-1 "doctor" will often take the same H&P, present to the same senior/attending, perform the same simple procedure under ultimately the same supervision, stick the same finger up the same... etc.
When I performed my first I&D, the surgical intern mistakenly introduced me as doctor. He verbally guided me through every step, which I thought the patient would find very awkward. But at the end, after all the screaming, the patient commented on how nice it was for young doctors training each other, and thanked us for the care we provided. On one hand, I could have clarified that I was just a student, my first month in the hospital, and most likely the patient would have asked the "doctor" to do it and for me to watch, and like most people I learn very little by watching. On the other hand, I did a good job and learned a valuable skill from which my future patients will benefit. The next time I was able to do it with minimal verbal guidance, just someone watching, and so on.
In another instance, I was walking with a different intern to remove a patient's staples in a postop wound check. "Hi, I'm Dr. X, and he's the student. Would you like him to remove the staples or should I?" The patient quickly said "you do it, he can watch." I've remove my share of staples, so while the intern was struggling (it was actually his first time), I was just watching the patient's TV. A few staples into the removal, patient looked at me and asked, "why aren't you watching him?"