There exist a cow in my barn
Well, this is more than an existential statement. In that it goes further than merely asserting the existance of a cow. It asserts that one can be found within the confines of the barn. This is a testable prediction because it is equivilent to saying: "if you go inside the barn, you will observe a cow."

Refuting the above statement does not refute the existence of the cow in question, it refutes the prediction that the cow can be found at the specified location.

Can you not see the distinction between these statements? Asserting the mere existance of a cow does not make any predictions. Asserting that a cow can be found at particular space/time coordinates is a prediction and can is therefore falsifiable.