You are reading too much into this. It's much more simple. Since the child lives with the mother, only the mother can claim the child. She does need a form from the father nor does she need to have any kind of custody agreement or any other paper work from the father.
It's only when the mother wants to let the father claim the child that form 8332 is required, from her to him.
The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332).
