I think it’s also important to make the distinction because there is a sizable number of children with Chinese parents who hold American passports. Yes, the children are American but that may just mean that their mother happened to live in the States on the day the child was born. Some of these children have not attended schools elsewhere, may or may not speak other languages, etc.
In my original post I meant this in a positive way. If you are looking for an opportunity for your child to meet and make friends with Chinese-speaking friends, you may be able to find that within the context of an international school.