To explain what I mean, I think you can level up a cooking style. For example, pasta. At level 1, you’re boiling dried pasta and adding sauce out of a jar. At level 1, you add your own spices. Level 3, switch to fresh pasta. Level 4, make your own sauce. And finally at level 5, make the pasta from scratch.
So with BBQ, I guess level 1 would be cooking the meat so it’s neither burnt nor underdone. Maybe level 2 is mixing different meats/cuts that have different heat/time requirements and cooking well. Further levels = ?
My part of the world uses the word “barbecue” specifically to mean the kind of cooking we were talking about and avoids that ambiguity you were struggling with, so I think I can confidently say use of the word “barbecue” as a synonym for “grill” when the word “grill” is also in use is uselessly defective.