There is indeed a normal contact force.

It sounds like what happened is your teacher wanted to model the free body diagram of the ring with gravity and two tension forces. This is the conventional way to calculate things like the tension in the strings. Adding a pair of normal contact forces to this free body diagram complicates your diagram without changing the answer. It's simpler to think the ring and the tiny bit of the string it touches as your "free body". In this model, the normal contact force is an internal part of your free body.

I'd say you and your teacher are both correct, from a certain point of view. You are correct in that there is a normal contact force. Your teacher is correct in that it shouldn't be drawn on the free body diagram.