The distribution will also give you a specific dollar value corresponding to that percentile, and then you can use the cost of living statistics to interpret it. If you say that 20% of households earn less than 35 k$ is that 35 big or small? The income distribution alone won't tell you that. In relation to the rest of the population, it's small, but what can you actually buy with that? Interpretation requires digging into the cost of living statistics. Once you combine the two, you'll be able to tell that the lowest 20% are struggling.

Percentile based classification is good because it is simple and it won't do the interpreting for you. If you can afford to eat, but you're not eating meat every week, is that really poverty or not? If you can travel, but you're not doing that driving a Ferrari or sitting on a private jet, are you really rich at that point? These are the kinds of questions the percentile approach avoids.

BTW the 2/3 median approach is really weird. The number of people it involves changes every year, but at least it's tied to a median which actually tells you something. Using the highest income as a benchmark makes even less sense, because that number changes radically every year based on what a few billionaires are doing. It won't tell you anything about the rest of the population. Using a wild number like that as a basis of any kind is pretty much useless. Besides, the highest income of the population is really far from the rest, because the distribution has an extremely long tail.