"...heated the house with coal fireplaces. The one in her living room had a water back of some kind for heating water, so if you wanted a hot bath or water for dishes, you had to light a fire."
That would be the 'back boiler'. Essentially a metal liner jacket containing pipework, fits into the fireplace, effectively becoming part of the grate (back wall and left and right sides - however, there is a hidden flue channel just behind the back wall. It is supplied with water, in a closed loop, heats the said water, which then returns it to the copper cylinder, usually unaided - but an electric pump could possibly be used - I suppose.
A damper-director lid (partly hidden up the chimney breast) controls the stream of burning 'gases'... when pushed fully in, the flames stay to the front, heating the room with a real blazing fire.
Damper pulled fully out, the heat mostly shoots up the hidden flue channel, boiling the water in the cylinder given half a chance. It is possible to wiggle the damper somewhere between the two extremes, giving a decent room heat and supplies of hot water at the same time.
I saw one example once, which had a rudimentary 1950's central heating system off the livingroom coal fired 'back boiler'. No pump was used. It used the principle of hot water rising to the bedroom radiators above. Not terribly effective.
And of course, an updated modern gas variant was/is the Baxi Bermuda type(s).
