Is pancake batter basically flour and water?

With some baking powder to make it rise and a little salt for flavor and it tastes better with buttermilk or soured milk instead of water and it also is fluffier that way. 

The traditional recipe basically consists of the combination of three ingredients: egg, milk, and wheat flour.

No, it needs baking powder or baking soda and cream of tartar too.

No

Its flour, baking powder and salt plus a liquid at minimum 

At my home its flour, baking powder, salt, eggs and milk (usually butter milk),

Flour, sugar, salt, baking powder, baking soda, eggs, butter, buttermilk.

Pancake batter must contain (besides the flour and water) baking powder, or it will not rise during cooking.  Other ingredients such as egg, milk, etc are options.

No, it should also contain egg. I do understand that vegans do make a type of pancake without egg, not sure what they use as a substitute, or what these vegan 'pancakes' are like.

No thats more deep fryer batter. The basic pancake batter is Flour, Milk and an Egg

Plus baking soda and eggs, try using ginger ale instead of water or milk.

No. Flour and water is basically paste. We used to mix water and flour, then heat it up until it becomes clear then we use it as paste if we don’t have a glue stick available. 

It can also be used as a batter for fish sticks or fish fillets or chicken nuggets. Pancake patter is water, flour, eggs, milk, and baking soda & powder, sugar, oil.