Poor design IMHO
The engine in the Cooper and Cooper S has an electric water pump. It only circulates coolant around the engine when the engine needs cooling - i.e. when a sensor tells the ECU that the coolant has reached a certain temperature in a certain part of the engine. There will likely be another separate sensor which control when the radiator fan starts spinning - more than likely, that sensor is different to the one which controls the water pump functioning, and is likely screwed directly into the radiator core rather than the engine block.
So on short trips, guess what. The water pump isn't switched on, because the coolant isn't hot enough to require it according to the ECU, but the sensor which starts up the radiator fan, says 'switch on the fan' because of the slow trickle of very hot coolant coming out of the cylinder head.
Seems to me that this would totally explain what people are seeing on short trips. On longer trips, when everything is in thermal equilibrium, then normal rules would apply, and the fan would only run when things get really hot. On short trips, the two sensors are saying different things, hence the strange behaviour of the fan.