My mother and step father got together when I was 6. I have always called him by his name when I talk to him but if I'm introducing him to friends or whatever I introduce him as my step-dad. My kids call him Pappy, they have never met my bio-dad and never will. I haven't even seen or spoken to him in about 10 years.
I am not with either of my kids' fathers anymore. They both see their dad's though. They call my DH by his name but he's said if they eventually wanted to call him dad he had no problem with that but its up to them. They are 5 and 6.
My son started introducing my now ex as dad... Sometimes he'll call him by that but the majority of the time, he calls him by his name. He knows that's not his real dad... He also refers to his real dad by his name, not 'dad'... His bio dad has never really been involved, and I was with my ex for 5 years.