Instead, my interest rests in the fundamental stakes of this system of classification from rules lite to rules heavy commonly found in our hobby. Why does this classification matter for some people?
Because people have preferences in regards to how much rules a game has and how much time it will take to reach "system mastery" using those rules.
What is gained through classifying games along this spectrum?
The same as classifying books, games, and media by genre. So the audience knows what to expect when they pick it up.
What is at stake if D&D 5e, for example, gets classified as Rules Heavy rather than Rules Medium?
It would skew the most common way to define what those categories mean. It would require we have new categories, something like "rules ultra-heavy" or similar for games that are clearly and wildly more involved, rules wise, than D&D 5E. 5E is generally considered rules medium, as it's the benchmark by which all other games are judged, that's typically a good thing. Rules light = less rules than 5E. Rules medium = about the same amount of rules as 5E. Rules heavy = more involved rules than 5E. Not the best system, obviously, but it passes the sniff test.
Are there incentives for games to be perceived and classified as lighter than they accurately are?
Lighter rules imply easier use, less to memorize, less flipping back and forth finding rules in the book, and less time to reach system mastery. Whether that's true or not is a debate for another thread.
For me, I prefer ultra-light rules. The lighter the better. If it can't fit on a page, it's too much. A paragraph or two, even better. A sentence or two, better still. So when something is described as rules light I'll pay attention to the rules themselves. Anything more involved than that, rules wise, might get me to check out the setting. And, if it's interesting enough, I'll use it with some rules light or ultra-light system.