What is with the sentence #2? I mean, why is it not so common?
- Hey! Stop splashing me! (in a pool)
- Hey! Stop splashing water on me!
because if you are in a pool (or the sea, or watering plants in the garden) it is obvious what you are splashing is water.
In fact 99.9% of the time you will be splashing water so the additional information is not required.
In a washroom:
I splashed my face with cold water.
I splashed cold water on my face.
In this example I am presuming you mean public toilets by the very coy phrase "washroom"
In these 2 examples the cold is the extra information, you could have splashed your face with hot water.
But it is also less common, it is more normal to use the toilet and wash your hands so splashing your face with water needs full explanation (well it does here in colder climates)