a site with a picture of the most common type cooker in small restaurants:
http://www.bigtray.com/catalog.asp!catid.10810.html a grill

unfortunately, the pictures of salamanders are not available. Hobart is one of the leading companies for industrial food equipment-- at least in NY there might be others in other part of the country.

a real restaurant, as opposed to a coffee shop/greasy spoon, might have something like this:
http://www.chefsupplies.com/chef/item.cfm?Section=Equipment&SubSection=Cooking&Category=Ranges&ID=790080

In US, we tend to have stoves, and or ranges, not cookers (and no one, but the very rich have anything like an AGA! ) http://geappliances.com/shop/prdct/ckng_gas/

It has become trendy to have Restaurant type stoves, (which most small coffee shops would not have) with 6 to 8 "Burners" and in some case the burners can be covered with a griddle, or a broiler. Like above, but with out the 2" side grill. They also have salamanders, a drawer under the burners for "cooler broiling" good for toasting the top of a meringue on a pie, or creme bruile. (since the heating source is above the food, and heat tends to rise, it is not as hot.)

Most US stove have a broiler area-- in electric stove it is the top oven heating element, in gas stoves its a drawer unit under the main oven. I have a "cook top" and separate wall oven with a "broiler drawer" below. (see ge site)

gas or electric would depend on the area. NY is privileged to have the highest KWH rates for electric power in the country. It is rare to find electric stoves in the area.. Where gas is not piped in, householders use bottled propane gas. In other part of the country, electric stove/ranges are the norm.

even BBQ-- you do that on a grill (often out of doors) but BBQ is it not just grilled food. a steak cooked on my grill is not BBQ'ed, its just grilled. BBQ food is also grilled, but , oh forget it.... BBQ mean something different in every state! i think Aussies use BBQ to do what we would call "grilling"
one characteristic of BBQ is sauce-- sometimes as marinade, sometimes as poaching liquid, sometimes added while cooking. the sauce can be sweet, or tomato-y, or sour or smoky, but its always a bit spicy. every region has a different type sauce, and a different time to add the sauce, and different food that is most often BBQ'ed.. pork, beef, chicken, seafood, lamb, turkey.. all are BBQ'ed

cooking (home cooking) doesn't seem like a specialize activity, but it seems there are different terms used in all parts of the world.
appliances:
Range/stove/cooker, hot tray, microwave, toaster oven, rotisserie, indoor electric grills, convection ovens...
pots:
sauce pans, saute pans, fry pans, pots, woks, bain maries, griddles, steamers, rice cookers, pressure cookers, electric kettles (just now becoming popular in US) gem pans (that one is almost archaic!) but M-W 10th knows what a gem is!
other--
so use a colander or sieve? when you use a spatula is it plastic or metal; if metal, shaped like a wooden tongue depressor or shaped like a shovel?

a recent house guest at my house couldn't figure out how to use my can opener-- and i have a simple hand cranked model...(standard issue in US)