I really try to avoid referring to watching my kids as "babysitting."
But it's hard to find a term for it. I hate to say "watching the kids" or "it's my day with the kids" because that also carries the connotation that I'm somehow only responsible for my kids occasionally. But "babysit" is the worst term by far because it seems to indicate that the man is doing a distasteful task for a little money (kind of like my day job at Burger King).
For the record, I don't "babysit" my kids. Ever. I made them. They are 50% me (scary as that might be). I am therefore fully responsible for the little munchkins, all the time.
And I'm happy to do it. That's the trade off that I signed up for when I became a parent. Many--maybe most--men don't want that responsibility all the time. They think that they are somehow entitled to sit around and drink beer while the woman chases the kids around. After all, they had a tough day at work and deserve the downtime. Right?
Anyone can be a father. But only someone special can be a daddy. The men who refer to watching their own children as "babysitting" only want to be fathers, not daddies. They are really missing out on something special!