She is only free at restaurants if she eats off your plate so you were correct in paying for her.
Just to clarify, the child does not have to literally "eat off your plate". At buffets they can get their own plates of food and eat free. Also, many places will bring out something small (a bowl of pasta, etc) at no charge. My sister intended to share all her meals with my niece, but most CMs offered to bring DN something on her own plate.
I would assume if you order a child's meal from the menu that you should/would have to pay for that meal.