I think the only issue is where the title is used. I remember a few years ago (more like 8) when I was hospitalized for foot surgery, I didn't know about any of this crap - but I was fully aware when I was seeing a Physical Therapist, the child psycologist to help me deal with the surgery, and the DPT. The grey area is when someone walks into the inpatient room - are they a doctor? a nurse? That was back in the day when part of me still believed that every male in the hospital was a doctor and every female was a nurse. For a nurse to go into a patient's room where the patient isn't really sure what to expect, and call him/herself "doctor", it is deceiving. I understand that NP's have a great deal of knowledge, but the truth is, they are not physicians. If it was you in the patient bed, how would you feel.
I know plenty of NPs who have introduced themselves as "Nurse Practitioner __." DNP's have earned a doctorate, but they should keep the title out of patients rooms. If I hear doctor, I think physician. If I were a nurse, at the nurses station or in nursing school and had someone introduce them self as "Dr. ___, but you can call me___", it would be understood. It's like Joe Schmo who earned his doctorate in Spanish Languange (yes, I know someone) - in the academic setting, the use of the title is appropriate. If he walked in to a hospital and introduced himself to a patient as Doctor, there would be an issue. I understand that is an exaggeration, but I really want people to understand the patient side of it (yes, sometimes it has to be about the patient, not just personal pride).