What exactly is a "moon"? Recently some new moons of Jupiter were discovered. However the moons are pretty damn small indeed - only a few kilometers wide. That makes me wonder if they should even be called moons in the first place. The Wikipedia article on natural satellites says there's no clear lower limit (or upper limit for that matter). But if we accept that, then Jupiter already has countless numbers of moons in its ring system, and even the Earth has multiple moons since the International Space Station is a "moon".

What exactly is a moon? Is it really a vague notion of "something that goes around a planet"? If so, why hasn't the IAU clarified what exactly a moon is? They've already done so for planets; it seems natural to do so for moons as well.
Unlike "planet" the IAU hasn't attempted to precisely define "moon". General usage requires that a "moon" is a natural satellite of a planet (or dwarf planet, asteroid, or perhaps even of another moon?) and it is big enough for us to have seen it as an independent body.

This contrasts with the "moonlets" that have been detected in Saturn's rings by their graviational effects on the even smaller ring particles. We know that a body exists there, but it is too small to be directly imaged. 

As such the question "How many moons does Jupiter have" is probably unanswerable, since it depends on the sensitivity of your telescope and how close you are to the planet.

The smallest moons are about 10km across, but moonlets in the rings exist that are only a few hundred metres across. So in current usage, 10k might be roughly where people stop saying "moon" and start saying "moonlet".

Language exists to serve us, the definition of planet was only made to solve a particular naming problem.  No such problem has arisen for moons. So there is no authoritative answer.