The dictionaries do not provide good enough answers for the right use of these prepositions in the following context:
Besides, which of one of these it is correct and why?
If you are in X, you are surrounded by X on all or most sides.
The surrounding can work vertically (I'm in a cardboard box) or horizontally (I'm in a box I drew on the ground with chalk).
If you are on X, X is touching your feet or the bottom of yourself, and X is a surface or something flat. If X is something flat and you are on it, you are generally on the top of it.
If you are under X, the "top" of X is generally above your head. At this point, if X is also surrounding you on all or most sides, you are also in x.
Being under X, if X is a surface or something flat, means it is on you, and you can't be on it.
Water is both something that can you can have the top of beneath your head, something that can surround you on most sides and is also something that is a flat surface so it works with all these prepositions.
So you can be:
He is under the water.
(source: ytimg.com)
He is below the water.
can be understood to use "water" as a layer which something is underneath
>The Abyssal Plain and Oceanic Crust is **below** the ocean
He is in the water.
has the meaning of being surrounded, in part or in whole by water. If you are under the water, you are also in the water. If part of you is above the water's surface, you are in the water.
Of your prepositons, one can only "Splash in the water".
on the water.
is usually used to describe a boat during an activity, since it is "on the surface of the water"
The boats are sailing on the water.
whereas
in the water
is usually used to describe a static position of a boat
>We took the boat from dry dock and put it **in** the water.
We launched the boat **into** the water.
