I can't speak (write ?) for all Psych nurses but I can for myself. I tend to listen a lot and I believe that helps someone in Psych Nursing so maybe I also tend to lurk instead of speaking out.
I've worked in Psych Nursing for 17 years as an RN and before that as a Tech/ Nsg Assistant. Initially I graduated from a Diploma program, got my BSN and then my Masters in Psych Nsg. I've worked with children, adolescents and adults of all ages. The patients I see seem to be sicker so a Med Surg background can only help. However the basic skills you learn in school are the ones you'll use and build on throughout your career.
I am cheap so I agree with letting your employer pay for education. That might be a benefit you look for when evaluating job offers.
And yes, there really is such a thing as a Nurse Psychotherapist. For years Psych Nurse CNS did this. Now many Psych nurses are becoming Nurse Practioneers and in many states they can RX medications. However they still use the basic and advanced Nursing and Psychiatric skills while treating patients.
Hope you continue your intrest in Psych Nsg and this reply answered some of your questions. If you have other questions please send me a message.