You are right, it doesn’t make sense. No matter what, the inner organs have to function at some level, producing heat which the body has to transfer to the surroundings, otherwise the inner temperature will rise. The main physiological mechanisms for losing heat are to increase the blood flow to the skin (vasodilation) and sweating (evaporation). And the body will try to use these mechanisms, no matter how futile it may be in a high-temperature/high-humidity environment.
Actually, neither does the suggested responses in low temperature make much sense. Again, the body has to preserve its inner temperature, this time by reducing the blood/heat-transfer to the skin (vasoconstriction), shivering and raising the feathers or fur. Maintaining the core temperature is paramount, and eventually, the skin and extremities will be sacrificed if needed. Numerous examples can be found!
There is however a small amount of truth in the hypothermic response: people adapted to cold environments can control bloodstream to different extremities (fingers or feet), avoiding frost-bites or for dumping excess body heat after heavy exercise, avoiding to sweat all over the body (Ref.: Schmidt-Nielsen, Animal physiology adaptation and environment, Cambridge University Press 1990).