The so-called key word "natural" is not a scientific concept. It is an arbitrary assignment of meaning that cannot be ascribed without indulging a circular argument or a tautology, such as "natural is whatever is not supernatural." But what ultimately determines whether a phenomenon is natural or supernatural? I think you'll find it is determined solely from the standpoint of each observer.
How do you know the force of gravity is not supernatural? Fact is, you don't. We've merely adopted a human convention of calling it "natural." Did it suddenly become "natural" just because we've seen it in operation for a while? Sorry, but nomenclature does not effect the nature inherent in a given object.
By insisting that the definition of science necessarily employ the arbitrary term of "natural," you have not only constrained it to your own narrow version of what you think science ought to be, but you have introduced a concept science cannot definitively address.