I always will be my children's dad/father but in public it's Izzy.

So what's wrong with that.

If someone is so wrapped up with semantics

As Jenniferinutah said, I think it's kind of an earned thing. My fiancee's "father figure" was her grandfather, as her biological dad was MIA and her mom hadn't remarried yet. So he will always and forever be "daddy" to her, and she calls her biological father and her stepfather by their first names...so on this topic, it's kinda hard to say.