Dear Father Steve: Apparently there was an uncommon word for a device to facilitate cooking in a hearth or fireplace:

4. A kitchen grate. [Obs.]

He was bid at his first coming to take off the
range, and let down the cinders. --L'Estrange.

Surprising that from being an uncommon word, it became common.

As for stove to cook on, the early kitchen stoves were triple purpose. The heated the room, pots and pans could be heated on top of them, and they had an oven for baking bread and other things. Remember the story of Hansel and Gretel, when the witch tried to get Hansel to get in the oven.

Cooker suggests to me separate devices, Pressure cooker, a large pot with a top with gasket and clamps to create hermetic seal, and pressure guage to set and control steam pressure. Very hand at high altitudes.
I see no reason to find fault with the UK term, and it is used almost exclusively to cook food.