-When the clutch is engaged whilst going down hill the wheels turn the engine. No Fuel Required. -When the clutch is disengaged the engine has to turn itself over - which needs fuel.
*Disclaimer, i'm sure carbourettas (sp?) could well be doing weird ass things so whilst conceptually the above holds they may well be different.
Ditto when coasting to a stoplight - with clutch disengaged (or in neutral), it would coast for a longer distance, which - if you're careful and know your vehicle - means you can let up on the gas a lot sooner.
Not calling you out specifically, but a lot of people arguing for manual transmissions in this thread seem to base their experiences on ~10 year old cars or older, and weirdly assuming that nothing has happened since, that software and technology in cars has been magically frozen in time, despite enormous incentives for car manufacturers to improve it.
To test, watch your rpm when you left off the gas, if it drops to ~750-1000 rpm, your torque converter is slipping (as designed) and not maintaining fuel cut.
To agree with you, I think the comments in this thread started from a manual transmission standpoint and then everybody thought it applied to automatics.
In a manual you are burning fuel when the clutch is disengaged to keep the engine turning over. With the engine engaged you steal from momentum which lowers fuel use.
The mid point of keeping the engine engaged and adding enough fuel to reach your desired stopping point once the RPM enters the same range as an idling engine is actually most efficient.
The oldest car I have that has an ECU is from 1987 (although judging the fuel efficiency on that car when it was new is a bit hard since the engine has been completely reworked since then -- including an aftermarket ECU, new injectors, new ignition etc which makes the car a tad more fuel efficient than my 2012 car)