I've noticed the phenomenon when making a bath too hot, but not any other time. Usually the water that feels OK on the hand feels far too hot when lowering in one's ..ehem... nether regions. But as you say, sometimes one experiences that confused feeling.

When preparing a bath for a baby, you are advised to test it with the tip of your elbow before risking the infant!

Your hands are very densely packed with receptors for hot, cold, pain, touch etc. and the rest of your body has variable amounts of sensitivity, but mainly relies on your hands for information. I'm guessing that the confusion is just that, a shriek sent to the brain but without detailed information - sort of, "Get me out and I'll tell you why later".

I'm trying to think of examples, but generally I reckon my hands always know hot from cold.

As for the temperature at which this happens: I imagine it would be hard to contrive a safe experiment to test it. You don't want your volunteers going home with 3rd degree burns, after all.