That’s a good point, and it’s something I always try to explain succinctly, and always fail.Let me add some complication. In the UK, grilling is done in the oven (AKA: cooker). It is what we Americans call broiling. What we call grilling, they call barbecuing.
There are two kinds of definitions for words/concepts - what the dictionary says, and what people actually practice.
Me, being a casual outside cook who doesn’t get too hung up on terminology, and as someone who sees very little black and white in the world, and a whole lot of grey, I say that anything I cook on my outside propane cooking apparatus is grilled. I’ll sometimes say it’s barbecued, but that usually means there’s some kind of dry rub or tomato-based sauce involved.
IOW, I make barbecued chicken (slathered in sauce) on my outside grill. I grill hamburgers on my outside grill. Doesn’t matter if it’s coals or propane, direct or indirect, low and slow or hot and fast.
If I’m inviting you over, I might say, “C’mon over Saturday, we’re grilling out,” or “cooking out,” or “having a cookout,” or “we’re barbecuing.” They all mean the same thing to me.