In this instance BBQ is referring to a specific type of food - that is, smoked meats, usually beef or pork, which is then covered with barbecue sauce, which is usually tomato-based. The specific meats, rubs, and especially sauces vary from place to place, and this is what people are referring to when they debate which state or region has better BBQ.
A barbecue could refer to an outdoor grill, but I don't think that's too common in the US. More often, it refers to a party held outdoors in which food is grilled, usually burgers and/or hot dogs.
You're thinking of a grill, which is used to make the BBQ. As for how it differs from state, I think it's the recipes for the BBQ that differ (some places use vinegar, others use ketchup).
Pretty sure they are comparing BBQ restaurants which are definitely a thing, and differ in style from state to state.
As you’ve implied, if you’re cooking the meat yourself in your backyard then you can do any style you want. I believe “true bbq” can be considered more time consuming than just throwing meat on a grill, and would involve the use of a smoker and perhaps other equipment. It’s not just grilling meat.
Different regions in the US have very different barbecue styles. There's a vast array of different ways to long-smoke meat over wood, and a vast array of ways to sauce and prepare it after it's been cooked.
Since good barbecue often takes years of experience, 16+ hours to cook, and produces more food than a small family can eat, it's a cuisine that is frequently outsourced to restaurants. So the kind of barbecue which dominates an area really has an impact on the kind of barbecue you can eat in that area, unless you've got years of experience, hours of spare time, and either a big enough family or enough freezer space to cook your own barbecue regularly.
(not sarcastic, I'm just a clueless foreigner)