No, you have not missed anything. You are parroting the "logic" of the Committee for Small Body Nomenclature of the International Astronomical Union. This is a true committee of fifteen members whose job it has been to decide on definitions of words. There was and is no science here. Nor was there any logic based on science; the logic was that of taxonomy: making pigeonholes to classify stuff. Nor was logic used in making the final determinations; what the pigeonholes were to be called was decided by vote. It was a "let's make new words" party, having nothing to do with astronomy, geology, or selenology. (See? It is both easy and fun to add words to the pseudoscientific jargon. Even scientists can do it!)
The Moon is considered a moon as the barycentre is within the Earth.
The barycenter of the Earth-Moon binary system (and that is a legitimate phrase) is always 1,000 miles below the lithosphere of the Earth, and 3,000 miles above the Earth's core. Quito, Equador, is a city on the equator. When there is a lunar eclipse on either the Spring or Autumn equinox at Quito, an interplanetary voyager arriving from Mars would find that Quito was 1,500 miles closer to the Sun than usual, but 12 hours earlier or later it was almost 1,500 miles further from the Sun than the navigator's first order approximation*. The communications officer of that interplanerary ship had better take into account the way the Earth spins about the barycenter of the binary system if he is to stay in laser beam contact with the Quito space port.
More significantly over the Earth's history is that its rotation around the barycenter raises tides. Not just the noticeable ones in the hydrophere, but large ones in the various layers of the atmosphere, and smaller, but significant, ones in the lithosphere. Geology has yet to develop an effective model on how the tidal strains on the lithosphere affect plate tectonics. But there can be little question that significant tidal forces are at work, alternately stretching and compressing faults.
In retrospect, what this august body of astronomers should probably have done is given their naming problem over to the experts who have recognized degrees in the appropriate field of study: these kinds of taxonomic decisions are better left to the linguists and other language experts. There are probably very few astronomers who have done any study of language arts at all. No wonder they bungled the thing so badly. They probably did not even know they were not doing astronomy any more.
I would not mind having someone check my geometry here. I think the difference is actually 3,000 miles (displacement of the Earth's center from the barycenter) but I'm going with the more conservative number.