Please read:
Lift-off oversteer - Wikipedia.
Also, if you have the means and opportunity, I recommend driving in a skid car (
SKIDCAR & SKIDTRUCK Systems: Complete Driver Training Solutions)
The reason most street car are tuned for varying degrees of understeer at the limit is the counter-intuitive nature of dealing with oversteer.
Lifting off the throttle is one of the worst things you can do if your car is already oversteering (braking is worse). Neutral, to slightly positive throttle is the right way to handle it in performance driving.
Note that drifting involves throttle too, but in that case, you're using *extreme* throttle to overpower the traction of the rear wheels and keep the rear sliding - that's a different situation.
In this instance, the right way to handle the situation was not to get into it in the first case (i.e., slow down).
BTW: I think hydroplaning is a probable mis-diagnosis of the situation. It was the bump in a turn that broke traction and started the car sliding. Once a car starts sliding, it doesn't stop sliding that easily, especially on wet pavement. This was simple over-driving.