I've always thought of computer science as a branch of mathematics. To me coding or programming is applied (or engineering) computer science, it's an engineering field (hence the term software engineer). Just like there is theoretical physics (a basic science), and also applied/engineering physics (e.g. mechanical engineering). They are related fields, but specializing in one does not mean you will be good at the other (it certainly helps).

The terms coding and computer science do get used interchangeably by a lot of people though, since most may not understand the difference. I've learned to live with it (occasionally I point out the difference, if I think the person misusing the terms actually cared to know).