You sound like you're a young student, and so I empathize with the embarrassment you must have felt when the professor called you out. I wish the professor had just let it slide, for your sake.
But now it's time for some self-awareness. I'm going to avoid entering a discussion about "pre-med narcissism" directly, because I anticipate that it will just devolve quickly.
If you want to get to the root of the problem, consider what "pre-med" implies. The label essentially says that you are pre--a-real-career. That you are just attending college because it's a hoop you have to jump through to get to medical school. It's very easy to refer to yourself as "pre-med," but the label really tends to turn people off because it oversimplifies what is, in reality, a long, complicated, arduous process. You should never go to college because "you have to in order to go to medical school." You shouldn't look at college as just a meaningless preparatory overture before the real show starts. College is so, so important for your growth, your maturation, your evolution: both as a student and as a human being. If you aren't getting that feeling right now, then I suggest you reconsider why you do the activities you do. You have to figure out what these classes, what this time means to you on a personal level, and it had better be more than just a stepping stone to medical school.
"Pre-medical" belittles the process, and is often negatively received. I have many stories from my own education where the biochem majors intent on graduate school would frown upon, scoff at, and ultimately deride my own ambition to attend medical school. There are many reasons for this attitude, so don't oversimplify their responses to "jealousy" either (as I once saw someone do).
Are pre-meds narcissistic? I sure hope not. After all, your dream is (or should be) rooted in altruism and compassion for the physical and mental well-being of others. But should you avoid calling yourself a "pre-med student"? Yes. I think so.
Also, your psychology classmate is being far too literal in their interpretation of the word. No one would be *****ic enough to diagnose you with a DSM disorder just because you said you're pre-med. Maybe you are narcissistic in real life, but you certainly don't come across that way to me. Don't take that particular friend too seriously.
And once again, I'm very sorry that you were embarrassed like that. I find that kind of behavior from a teacher, a mentor, an instructor entirely unprofessional and unbecoming. That professor would get a scathing ratemyprof review if I had been in that lecture 😛
Best of luck to you!
EDIT: Wolverine's final pro-tip: When people ask me what I'm studying, I say, "Oh, I'm majoring in (e.g.) Psychology and (e.g.) Biochemistry, minoring in nobody-cares, but my real hope is to attend medical school someday. In framing my studies this way, I don't reduce the major to a stepping stone in the same way that you might have (accidentally, I'm sure) in saying "pre-med biology." Tell people what you're studying, have a great reason for studying it, and only after establishing that should you slip in "oh btdubbs I also really wanna go to medical school."