So this is super late, but for future students reading this post I'm an med student who came in to medical school with a PhD and had it on my 3rd/4th year white coat and it was fine. It didn't rub people the wrong way because I'm not an egotistical jerk who rubs people the wrong way. I agree with the poster above who said that 'being a thorough, genuine, prepared med student' is so much more important. For classmates (to my knowledge), it never bothered anyone because during pre-clinical courses, when my degree was relevant, I was a helpful peer-tutor, so it benefitted a bunch of people. For attendings who were basic science inclined or just noticed it, it piqued their interest and it was a great conversation starter, and I felt like it let them get to know me more as an actual person. Same thing for patients who were researchers or just friendly & talkative. I never introduced myself as 'Dr', because that would be intentionally misleading and inappropriate in a clinical setting, and if a patient ever asked 'oh should I call you Dr?' my response would be a joking 'oh no, only my mom calls me that 😉 ' (which is true)
Getting a PhD takes blood, sweat, and tears. It can be a rewarding, but it is often an isolating and emotionally tolling experience. Even if you don't plan on continuing research, it'll always be part of who you are. For me, it was part of my identity, it helped me get into medical school in the first place, and ya know, I'm pretty darn proud of it.