I've worked in psych all of my career, and have always had my first and last name on my badge, and introduced myself to clients with my first and last name. Only one employer in my career had a firm policy that I wasn't allowed
to have my last name on the badge (I complained about it, but got nowhere), and I made an extra point at that facility of always telling
clients my last name. I've lived and worked in small town rural areas and medium- and big-sized cities in a few different states, and it's never been a problem (so far!
Plenty of people with riskier jobs than nurses use their last names freely at work, and I never understand what the big deal is for so many nurses. While I understand that safety is certainly a concern (I am a single woman who lives alone), of all the things that one can do to promote one's safety, keeping your last name a secret from clients at work seems so far down the list to me that it's hardly worth mentioning. Clients who really want to find out your last name can do so pretty easily regardless of whether it's on your badge or not (they can just request a copy of their record, which has all our last names all over it. They can watch for you to come out from work (after they're discharged) and follow you home. Etc., etc., etc. ...
Of course, I think the whole business of going by our first names at work is a big mistake, but that's a different conversation.