I watched that and, while better than the current system, I still disagree with the proposed new definition. I think it goes too far the other way. The goal is to convey information in a way that is quickly and easily understood, clumping planets and moons together just complicates the process IMO, why would you want someone to think that Titan is a planet with the historical presumption that planets orbit around a star independently? Then there has to be secondary question about whether it orbits a parent planet or not.
In my mind it is simple: if it orbits a star and looks mostly round, you know, like a planet, it's a planet. If then there is another body that orbits around that planet, then that's a moon - I know, tricky stuff so far. If two celestial bodies orbit each other then they're both planets in a binary planetary system. If there's two celestial bodies that orbit each other but both orbit a planet, then it's a binary moon system. Etc..
At this point you can create a size scale, for planets; 1-x - x representing any size from Ceres to Jupiter and beyond. For Moons; M1-Mx - again x representing the same as for planets, but likely not to get as high up in numbers for obvious reasons. These can be based on percentage ratios with Earth's size as the starting point for both planets and moons, Earth = 0, smaller is -1 etc, bigger is +1 etc.. Work out a good mathematical basis that you don't need a supercomputer to understand, ie; that allows someone get a sense of scale of what a +5 planet or a -3 moon is in their mind. Then add any other discernible characteristics and you can adequately describe any object that is round(ish) in any system in one go. If you want to only consider the major planets, then slide the scale accordingly to find the type you are looking for, if you want to only consider the minor planets, do the same the other way. Repeat for moons.
I don't know, maybe the above is too simplistic and doesn't make one feel smart enough if one understands the differences compared to those who don't.