Not at all.
You bake something in an oven. It's a box that's heated.
A toaster has heated elements very close to what is being toasted. You toast bread in a toaster to make toast.
You grill by placing your food on a metal rack over heat. It can be a gas grill or a charcoal grill. You do this outside, usually, though a few fancy cook stoves have small indoor grills.
Under the heating elements would be broiling here.
(I agree on the toasted nuts thing - it just means subjecting them to dry heat, which you can do in a frying pan on top of the stove.)
No, our ovens usually have a small area under them with a broiling pan. It's a suitable alternative to grilling - the heat comes from one side, just like in a grill, and you flip the meat so get the heat to both sides, just like a grill. I've broiled hamburgers, fish, pork chops, the so-called "London broil" etc in my broiler.
Also, my little toaster oven - basically a small oven that you can either heat up like an oven, or turn on the elements just on top to broil - is prefect for melting the cheese on top of a piece of bread you've already toasted.
(So while we're on the subject, what is Welsh Rarebit?)