Taking a cold shower lowers your body temperature, then your hypothalamus kicks in to compensate. You get out of your cold shower, and your body heats up. If you take a warm shower, you end up cooling down after the shower.

Not my experience. Sometimes I take a cold bath (ice is involved) after running. With enough ice, I feel quite cold for many hours after getting out -- to the point where I get out an extra down comforter to sleep with, and am still cold.

That means you have lowered your core temperature and your body is simply unable to respond to that coldness quickly. It's vital that you feel cold at that point. Quick cold showers are just "tricking" your body.