When I was growing up in Southern California and we fired up the coals and cooked hamburgers outside, we called that barbecue. As an adult, particularly in my Oklahoma years, barbecue was when you marinated the meat in a secret family or restaurant sauce and smoked it over low heat for several hours. What we did when I was a kid was called grilling.
Somehow that distinction still stands for me. Today we have a few barbecue places near us by terms of the second definition. When we cook burgers (meat or veggie) outside at home (now using propane) I call it grilling.
Interesting how our use of language changes.