How to say it properly?
Don't splash water on me! / Don't splash me with water! / Don't splash me! (if it's clear that water is meant)
If the water didn't reach you because you were far enough, you can say:
Don't splash water at me!
As for the verb to wet, here is an example of how it can be used when talking about babies/young children:
The baby has wetted/wet the bed (urinated in bed).
So, you are right. Your sister really doesn't mean "Don't wet me!"