Should only spherical satellites be considered 'moons'?
The need to distinguish between these potato-shaped and spherical moons hasn't arisen. As such, there aren't two different words to designate these two types of moons. And as long as the community of astronomers who study natural satellites doesn't feel the need for there to be an extra category, there likely won't be one created preemptively. 

Perhaps, one day, we will discover that potato-shape and spherical satellites form in very different ways, of have very different properties (not just esthetic). Then the case might be made to sort them into two different categories. 

On a side-note, defining arbitrary categories is never trivial. How spherical is spherical? The boundary has to be set somewhere, but there always will be edge cases where an object could reasonably fall in either category.