I have told multiple dads that, it is not babysitting if they are your kids, then it is just being a dad.

My wife used to sau stuff like "you're on baby duty" or "I need to go out, so you'll have to babysit". She says those things because that's the way people talk. I pointed out to her every time that I'm the father, not the babysitter. She has changed the way she says those things now.