Would an analogue of the definition for planets also work for moons? This is a follow up question to What exactly is a "moon"? The conclusions I draw from James K's answer is that the IAU should define what a moon is. They haven't done so yet, but they should.

The obvious way to go about defining "moon" would be to build on the definition of a planet:


Planets must be in orbit around the Sun
Planets must be round due to their own gravity
Planets must have "cleared the neighbourhood"


This immediately suggests the equivalent definition of a moon:


Moons must be in orbit around the host planet
Moons must have [minimum size]. Unfortunately being round doesn't work since not all moons are round (c.f. Mars' moons, unless one reclassifies those as not moons)
Moons must have "cleared the neighbourhood" of their own orbit, i.e. in their particular orbit around the host they are by far the most massive body


Would such a definition work? If so, why hasn't the IAU also defined moons? It seems so natural after all. If not, what's the catch?
In 2006 the IAU had a trilemma. 


They could decide that Eris was a planet, and potentially allow for future discoveries of tens of new planets. 
They could be inconsistent, declare that Pluto was a planet, but Eris (and Ceres) wasn't
They could come up with a definition of "planet" that would exclude Eris, and consequently also exclude Pluto.


Each option is problematic: We teach eight-year-olds to chant the names of the planets, would we want to make them chant the names of 20 or thirty Kuiper belt objects?  Being willfully inconsistent would lead to further argument. Re-describing Pluto would go in the face of 75 years of tradition. There was no option that would keep everyone happy, but they chose the third. 

We don't have a problem with a more relaxed defintion of "moon" we are used to new "moons" being discovered. Eight-year-olds don't need to chant their names.  The casual definition of "if you find it, you can name it" works. 

The issues with planets are particular to our expectations that "planets" are important. We have no such expectation with moons.