Fluff time: is it chocolate milk, or is it just chocolate flavored milk?

I was at a diner eating with friends and I ordered chocolate milk, the waiter informed me that it was milk with a chocolate syrup additive. That got me thinking, how and why chocolate milk is chocolate milk and if you have chocolate flavoring like Nesquik, Ovaltine, Hershey’s chocolate syrup, etc. would that make it chocolate milk or just chocolate flavored milk? What is the difference, didn’t they have to add some type of chocolate flavor to the milk before they bottled it or sealed it in a carton?