Child custody in nc

I haven’t filed for sepration yet but I am about to do it. My husband cheated on me 2 yrs ago and has done it again.
I want to know what I will need to fight for in order to have child custody for my two little girls that would entitle me to travel with them or bring them back to my country whenever I want? (in a temporary or permanent basis)
is it only “Physical custody” or “primary physical custody”? or any other one??

My husband will have open doors to visit them, spend time with them, etc… but I want to make sure I understand this prior to getting into any discussions with lawyers or himself.

Again, you will have to have your spouse’s agreement to leave the country to reside elsewhere with the children, or prove to the court that it is in their best interests to live abroad with you.