Grilling doesn't seem to have a broad definition. Most people tend to agree that grilling means cooking meat over direct fire, which is usually considered hot and fast.
Barbecue is where the variations come into play. I hear a lot of my friends in the New England states refer to "The Barbecue" as the device on which they are cooking. They cook their meat on "The Barbecue." They just don't refer to it as a grill. I also hear this used a lot from my friends in Australia and the UK.
Those of use here in the US who are serious about "Barbecue" tend to use it as a term to describe how our meat was cooked in an adjective form such as barbecued pork or beef. We are describing our low and slow process of cooking the meat over lower heat with added smoke. We use grills and smokers to produce our barbecue rather than a 'barbecue'.
As far as low and slow vs hot and fast is concerned, you guys can argue about that all you want. I know there is a huge faction of barbecue cooks who swear by 225° on their thermometers, which is fine. 225° produces excellent barbecue and I have cooked a ton of meat at that temp myself. Through a lot of experimentation with pork butts and brisket, I can't tell much difference between meat cooked at 225° and 300°. Some of you may be aware that I recently cooked a boston butt at 350+°, which also turned out very well. I saw some minor variations in that particular meat that I'm not quite ready to discuss until I can try it again.
So, if I had to draw the line between what is low and slow and what is hot and fast, I'd probably put the mark somewhere between 300-325°Anything below 325° would be considered low and slow while anything above 300° would be considered hot and fast. Yes, there is a 25° overlap there that I would consider some sort of no-man's land of sorts.
In my humble redneck southern US opinion, barbecue is all about experimentation. Don't even bother discussing the ramifications of low and slow vs hot and fast until you have 5 or 6 cooks in each realm under your belt! You can't put your money where your mouth is until you have tried it!
Don't let terms and debates like this separate you from what YOU believe each of them to be