In California, there are several aspects to the care and control of your child.
The first is custody, and there are two kinds: physical and legal, and for each there are two options – joint and sole custody. Physical custody is where your child lives. If your child lives with you and your ex, then you will have joint physical custody, even if one parent just has one or two nights a month. Only if one parent has a very limited amount of time with your child will one parent have sole physical custody.
Many parenting plans (custody and visitation orders) have a designation of “primary custodial parent,” which is commonly referred to as the parent that has more time than the other. Some parents are adamant that they want this designation. Legally, however, there is no significance to this term. It means nothing, and in my opinion is a potential point of contention that should be eliminated in agreements and orders.
The other aspect is legal custody. Legal custody is the responsibility for the decisions regarding the health, welfare, and education of your child. In the vast majority of cases, this will be joint instead of sole custody, unless, again, one parent is simply absent from the child’s life. Legal custody means that you have to make joint decisions with your co-parent regarding your child’s education (public versus private school? Religious training?), health (surgery? braces?), and can even encompass things such as haircuts (shaved heads and spiked designs come to mind), piercings (sometimes ears, but more often eyebrows and belly buttons), and tattoos.
A common desire by some parents is to simply eliminate them from their child’s life much as they are eliminating the other parent from their life. This is not likely to happen. If a parent wants to be involved, even intimately involved, with their child, then that is to be encouraged. It is understandable that one parent may want to put some distance between themselves and their ex, but the legal reality is that if you have a child with someone, then your life is going to be entangled with the other parent until that child is an adult, and often beyond that.