What a fascinating discussion.
Doesn't it show how words mean different things in different places.
As Aurora's links explain, broiling and grilling are basically the same except for where the flame is. With grilling, the flame is below the food. With broiling, the flame is above the food.
Ermm! Not in Europe. Grilling in Europe is done UNDER the heat. Stoves (electric and gas) often have a grill like a shelf under which you put a pan to grill the food. Most electric ovens (and many gas ovens) have a grill element at the TOP pf the oven and food to be grilled is put underneath. ye fat does not drip onto the heating element. We never use the word broil, assuming this to be an American term which we understand to mean grill under heat.
You can also buy gadgets which have a vertical heat source and food (especially chickens) are grilled on a horizontal rotisserie in front of the heat.
But here are also vertical heat sources where the rotisserie is vertical. This is a popular way of cooking lamb slices pressed together on the vertical spit.
And a tandoori oven has the food suspended vertically and the radiated heat is all around the food. Peking Ducks are cooked this was (in Peking/Beijing). I don't know whether this is grilling or broiling!