The coolant temperature warning light is controlled by a separate switch from the one that does the fan.
It's also separate from the ignition unit so there is no way for it to come on for a few seconds when you turn on the ignition.
The only time I've had an engine overheat was when the radiator developed a leak and the coolant level dropped. The switch is near the top of the thermostat block so if the level is low the switch doesn't get hot enough to activate anyway.
The only time it has come on for me has been in heavy rain.