It could be a number of things.
When you mean went through over a litre of water, did you just use plain water and not coolant? This often burns off as whie smoke through the exhaust if you use water, as condensation, and really you should stick with the correct mix of coolant to prevent the engine from over heating.
You could have a number of problems, a leak from the radiator, a blocked radiator, a stuck thermostat, or there could be potential head gasket or heater matrix problems.
First check all the pipes of the coolant system, are they perished, any leaks?
Second take of the oil cap and header tank cap and check whether there is any coffee/white residue which could indicate head gasket failure. Is there any smoke coming from the exhaust? If so what colour? Does the car run rough, does it idle crazy and misfire? This usually is the sign of headgaskdt failure.
I think you should flush the system, when was the coolant or the car last serviced/changed? Buy some rad flush, put it in the coolant system give it a flush through, and clean with a hosepipe on top rad hose and take off bottom rad hose. Then reattach hoses, and fill with the proper mix of coolant, usually red. Make sure you bleed the air out of the coolant system as this can cause over heating, usually you use the bleed screw/valve, and squeeze the pipes to move along any air.
You mention the heating in the car going from hot to cold, usually if there is no water in the heating will not work in the car, however if there is a leak, or blockage water will not be passing near the heater matrix, or that could be leaking. Is your passenger side carpet wet/damp?
I'd say you can check the following and do that for about £5 for the Radflush, if you have leaks people use Radweld.
Is your fan also kicking in when the vehicle reaches over operating temperature?