You're right stile, the 'grill' is the metal grate that holds the food. But, because we americans have gotten lazy, we have generalized the WHOLE concept and called the whole thing a 'Grill'. It even goes farther ...
"I want to grill some ribs."
"Im going to throw a burger on the grill"
In both the cases above the word grill is not referring to the metal 'grill' that holds the food. Even farther still the first example the word 'grill' is a verb, and grilling is an action. The second example has the word grill as a noun, but grill here is synonomous with the WHOLE cooking contraption.
if that weren't bad enough ... the word Barbeque (BBQ) has gone the same route. Traditionally BBQ was a type of cooking, characteristic of a sauce or flavors. BUT because you cook BBQ food on a "grill" BBQ now is also used as a verb and a displaced noun.
"I want to Barbeque today, maybe some steak."
"Im going to light up the Barbeque, food should be ready in 15 minutes"
Hopefully my examples were clear ... in all cases it makes little sense.