Am I wrong or is my coworker wrong? What should I do?

You are wrong for clocking in and then leaving. If you get hurt on company time, then workman's compensation is involved. It' s a legal thing. The fact that you work thru lunch and stay late is not the issue. You do not get to make your own schedule. Put a positive spin on your co-worker's "nosiness" and take her advise. Get your car parked and arrive to work on time every day.