This is a good point and also worth pointing out (in the US at least) that most ophthalmologists will be covered under your medical insurance not your eye insurance (if any) because they are medical doctors.

I was getting my contact lens exam/fitting recently and the optometrist wanted me to get something checked later by an ophthalmologist. He said it was probably nothing to worry about but made it clear that it was outside his scope of expertise to diagnose or make medical recommendations.