This is a great question. When speaking with others in the profession, I usually use externship as that is the common term that everyone knows. However, when speaking with those outside the profession, I usually refer to it as the clinical residency or internship year .
When speaking to patients I would usually try to avoid 'student' as much as possible. I don't want to mis-represent myself, but I also don't want them to think that I'm some noob undergrad. I would usually introduce myself as Chris and say I was a doctoral intern or resident. People are more familiar with this concept (and comfortable) than the idea of being seen by a 'student'. The doctoral part would usually make them feel more at ease.
Again, don't mis-represent yourself, but give them the confidence to know that they're in good hands (even if they're not.😉)