"Is it OK for me to refer to you all as "nurse anesthesiologists"? Because it sounds like since "anesthesiologist" is not protected, that it is OK and even appropriate. Am I understanding correctly?"
Hard to tell, some will like it, and some CRNAs may not. CRNAs that work for physicians are likely to get push back from them so are more likely to be uncomfortable with it.
I had an 'Aha' moment when reading about this over the past few months when I realized that many people don't think that 'anesthesiologist' means physician. Through my career I've been referred to as an 'anesthesiologist' several times by people who clearly were aware that I am a nurse. Then I realized they didn't see 'nurse' and 'anesthesiologist' as mutually exclusive, they think I can be both.
The most important reason to support 'nurse anesthesiologist' is the systematic campaign to have AA = CRNA. AA is now CAA, and they identify themselves as 'anesthetist', and policy makers are told AA = CRNA. So we must clearly communicate that this is not accurate. In some practices CRNAs are limited to the scope/function of the AAs, but the professions are different. We want everyone to know we are 'nurses' and 'anesthesiologists'. There are physician anesthesiologists, dental anesthesiologists, veterinarian anesthesiologists and nurse anesthesiologists. This is not about CRNAs wanting to be 'physicians' it's about CRNAs wanting to be recognized as nurses who specialized in the advanced practice of anesthesia.