That condition is due to the coolant sensor being overly picky about the level in the tank. Filling the coolant to be about an inch or so above the MAX line on the tank should resolve the problem. The light on the dash has nothing to do with the pump and will only light when the level of fluid drops. I can say from experience that if your pump goes, you won't know it until the coolant is gone and
then the light goes on
Still, it would be a really good idea to change the pump, which in my opinion is very easy to do, especially considering I'm not all that great of a mechanic. Simply clear enough room around the pump, remove the belt, unbolt it, reverse for the new one.
