Don't take it personally. If a doctor stepped in and said, "I'm a doctor, how can I help?" I'd keep giving orders and ask for his/her assistance. If a doctor stepped in and starting telling me how to run the scene, then I would say, "Ok, this is my patient, but you are now implying that you wish for them to be your patient by giving me orders. So, if you want to assume responsibility for the patient, you can take them as your patient, escort them to the hospital and I will assist your patient however you wish." Frankly, I'd love to have an emergency room doctor step in and assist. On the other hand, a podiatrist may not be up to current emergency standards, and it would be hard for them to justify taking a patient from EMS. In that sense, I don't take issue with how a doctor assists on scene, it's jut that I want to know what type of medicine they actually practice and who they really are. When I lived in Guatemala, a "doctor" from the U.S. was assisting at clinics in the jungle...until we found out that his medical license had been revoked 10 years ago.
So...do you keep your EMR card on you? Even if you do, how does someone that doesn't know you know that you can perform the skills adequately?
The appropriate response in the future is to calmly walk up and say, "I'm a trained emergency medical responder, do you need assistance?" If they say yes, help. If they say no, then walk off and leave the situation to them.