So, this question came up while one of my students was describing his own picture; in the picture , he was holding a fan made of feathers and was cooking the meat on the barbecue grille. So, I asked him," what were you doing?".
The answer should be: "Iwas having a barbecue."?
That might be a reasonable answer, but it's not particularly specific. "Having a barbecue" means roughly the same thing as "having a dinner;" it means to cook and eat food, while socializing with guests. If you're talking specifically about the cooking process, you can just use the verb "barbecue," as in, "I was barbecuing a steak." Also note that there are some who insist on making a distinction between cooking meat on a grill (or grilling) and barbecuing. When barbecuing, one cooks with low, indirect heat; grilling involves direct, higher heat. If that fan was being used to stoke the fire, it probably wasn't barbecue.
I hate to complicate things for you, emanko, but this is a little more complicated than I thought at first, and I am pretty sure there are some significant regional differences.
For me, at least, "Have a barbecue" doesn't just refer to the method of cooking. For me (and I know I'm not the only one), "a barbecue" is a an event - a party, really - that involves cooking outdoors.
So if I decided to cook out on the beach with just me, or perhaps just me and my husband, I wouldn't call that "having a barbecue." I'd call that "cooking out on the beach." If we had another couple over for dinner and decided to cook out on the beach, that might qualify as "having a barbecue," and if we had a party out there, that would definitely qualify.
But if it's just one or two people on the beach (or wherever) and cooking their food outdoors, I wouldn't say "I was having a barbecue." I'd say "I was cooking out on the beach."
It's also possible to use barbecue as a verb, e.g., "I was barbecuing." I don't say this, but I've heard it fairly often.