Should only spherical satellites be considered 'moons'? Since Titan and Ganymede fall into the same category as Deimos and Phobos or the components of planetary rings, the category of moons, it's like if all asteroids were considered planets. Tiny irregular satellites are called 'moonlets' but still fall into the category of moons.

Is there any distinction between the natural satellites and should there be one? From my point of view, only equilibrium-shaped spherical moons should be considered moons. While natural satellites are all satellites, the spherical ones should be moons. Otherwise it would be extremely weird that Deimos and Phobos fall into the same category as our Moon or the Galilean Moons. For example, Pluto should be considered having five natural satellites, and one of them (Charon) is a moon.
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.