Adding baby to insurance and hospital billing with insurance - help!

This is for people with private insurance like through an employer or independently bought -

Ok so I’ve heard the baby can’t be added to insurance until after they are born, makes sense. BUT I am confused… the baby’s initial hospital bill after being born - does that get billed under me, or will it be like a whole new deductible and out of pocket max just for the baby?? I know that once he is going to the pediatrician out of the hospital etc, all that will be under his part of the plan and his individual deductible will need to be met regardless of how much we already paid for me this year…. But I’ve heard that sometimes the baby’s initial birth costs at the hospital just for the “labor and delivery” stay come under mom’s costs and her deductible/ out of pocket max.

How did this work for you??? I have United Healthcare if anyone has any specific knowledge or experience with it all. I’m just trying to figure out if we should expect that the worst case is my out of pocket max, OR if we should expect to pay that PLUS the baby’s whole deductible and more? And I know it’s likely different insurances do it differently so I will end up looking into it, but it would be great to hear everyone’s experience first since I waited FOREVER on the phone twice already to get through to someone at our insurance 🙄

THANKS for any input!!