I've kind of struggled with this same issue. My kids still both call me dad, though my son at times switches it to 'D'. When I write or text him I usually sign it as such. He's in H.S. With my daughter, it seems I vacillate between dad and Julie. She is 21.
Recently, I was at my daughter's house where she and her significant other were hosting a party. To his relatives, I was introduced as Julie by her and her BF. This was fine, though at one point at the dinner table they each started to let the 'D' word slip out. I took it in stride.

My son's friends all know about me, and I have to say the funniest occurrence happened with my son's friend Bill. I've known Bill since he was at tall as my waist. I was also his soccer coach for a year or two. Bill's a great, smart kid with a good sense of humor. I hadn't seen him in a few years.
The story goes like this. Two years ago when the boys were 16, I was to pick up my son and Bill for a quick trip to Walmart for a video game. At this point, my son had just started seeing me again after almost a year hiatus while he was adjusting to my transition. I was thrilled that he had invited me not only to go out, but to have his friend join us.
As Bill and my son were getting into the back seat of my car, Bill let's this one out. "Hey, great to see you Mr. __________! How's it going? Bill and I commenced to kind of catch up on things and old times while my son sat in the car bewildered. Bill's nonchalant attitude helped to put my son at ease and showed him how this can be a non issue with people. Of course, while my son was scoping the games at Walmart, he excitedly called me dad a few times while showing me a game. That registered a few strange looks in the aisles!
So to your point Erocse, I guess with the name thing, I just let people kind of take it there themselves. Sure, when my kids were younger, I asked that they not call me dad while I was addressing the waitress or clerk, but otherwise I haven't put the reins on people.