Why would I be told not to give a two-month-old preemie any water at all, not even sterilized?
The answer doesn't just have to do with germs that might be fed to her with the water, so it's not the sterilization that is the issue.
The main reason we don't recommend water to two-month-olds or preemies is that the energy expended by such a small infant in eating should be rewarded with liquid, calories and nutrients, not just liquid. Breastmilk and all standard formulas have 20 calories per ounce, water has none. Drinking water will fill her transiently, but it takes energy to drink; energy she's better off spending getting calories.
For older, physically larger infants water isn't recommended either, at least for the first six months or so. Even on the hottest days, extra breast milk or formula provides the needed extra fluid.
Will water actually harm a child? No, as long as it is in addition to his/her regular milk intake and doesn't replace it. A mentor of mine once told me to think of water in your situation as a Snickers Bar -- OK every once in a while, but always in addition to a healthy diet, not instead of one.