(1) "A Macaroni French Cook," London, 1772.
It looks like this man is working at an inbuilt brick cooking hearth, with a fire at the bottom and a hole (or holes) at the top to contain the saucepan. It was a close relative of the French device called a "potagere" (or, in German, suppenherd, soup hearth), which enabled the cook to carry out operations relying on the indirect application of heat away from the main cooking fire, and was usually referred to in English as a "stew (or stewing) stove." It probably supplied some of the inspiration for the "Rumford apparatus" briefly popular in some upper-class American kitchens in the early 19th century, with its array of cooking holes each provided with their own separate, controllable fire. A "stove" of the primitive type illustrated, rather than its more sophisticated Rumfordized descendant, is the thing referred to in most early 19th century cookbooks, in Britain and America, on the fairly infrequent occasions when they mentioned stoves at all. Many early American cookbooks were notably conservative, possibly because they were created by copying from older (which usually meant British) exemplars, and continued to describe open-fire cookery with the occasional assistance of a "stove" like this, decades after American consumers up and down the East Coast had begun to adopt something entirely different, a cast-iron device of the kind that probably comes to mind with most people nowadays when they read the word "stove."
Moral: the meaning of words changes over time, and across national or cultural boundaries. This is obvious, but in this case it means that if we read references to "stoves" in late C18th - early C19th texts, particularly from the British Isles, we shouldn't assume that they mean "stoves" of the kind C19th Americans would have recognized.
(2) James Gillray, "The Comforts of a Rumford Stove," London, 1800.
This is a picture of an efficient, Rumfordized fireplace, scientifically redesigned to draw better, smoke less, and at the same time send less of its heat up the chimney. A coal fire burns on its grate. This is what English people usually called a "stove grate" rather than a stove, as the caption suggests; it became the principal method of space heating in 19th-century British households.
(3) "The Whim," London, 1808.
This shows a different kind of stove grate, a free-standing device made of cast iron and basically a descendant of the Anglicized form of the Franklin stove on sale in Britain from the late 18th century. The most obvious changes from the American original reflected its adaptation to a market where coal (requiring a raised grate, rather than burning on a bed of its own ashes like wood) was the main fuel source, and customers demanded a sight of the fire. In this picture, the poker sticking through the grate front into the coals is probably being warmed up to red heat to mull the drinks. [See James Sharp, An Account of the Principle and Effects of the Air Stove-Grates (Which Warm Rooms, &c. by a continual Introduction and Exchange of dry fresh Air,) commonly known by the name of AMERICAN STOVES: Together with a Description of the Late Additions and Improvements Made to Them (London: 1781, 6th.ed.) -- a 16 pp. promotional pamphlet, available via ECCO.]
[to be continued, maybe: with
-- an account of the final distinctive British device, the cast-iron kitchen range with an open fire between the oven on one side and a water-boiler on the other, that took the place occupied in American kitchens from the 1810s onward by a stove or range with a closed firebox
-- some reflections on what all of this signifies, i.e. how does it contribute to answering questions about why the country with the best-developed iron industry, already well advanced down the path toward dependence on coal as its principal heating fuel, did not adopt the much more efficient devices for cooking and heating ("stoves") that became ubiquitous in early 19th-century America]