Where I'm from, when you invite someone over for a cookout, it means you'll be eating outside but may eat a variety of foods - sandwiches, fried chicken, pastas, casseroles, fruit, chips, etc. and is usually very potluck.
When invited to a BBQ, it means you'll also be eating outside and will be served grilled meats. Grilled chicken, BBQ, steak, kebobs, burgers, dogs, etc. Usually, at least in my family, when you host a BBQ, you provide all foods on your own. There's no BYO/potluck nature involved.
For example, we had a cookout for Easter. I hosted and provided all drinks, plates, utensils, napkins, cups, a ham, a turkey, and a pinwheel sandwich tray from the grocery store. Other guests brought sides, desserts, chips, fruit, etc.
I'm having a BBQ for Mother's Day, but next Saturday, not on Mother's Day. I'm serving grilled BBQ chicken, kebobs, grilled shrimp, salad, veggies & dip, fruit, some pasta salads, and a cake I've ordered. Guests are only showing up to eat and enjoy each other's company. They don't need to bring anything and those who have asked have been told nothing is needed, just their presence.