That is definitely what the GP thread meant, but the terminology was wrong. They wanted "percentile" median specifically refers to the 50th percentile. You could also use "tertile" (e.g. "The second tertile has income between 30k and 65k) which refers specifically to spliting into thirds.
They are also the 2nd and 4th sextiles, on either side of the median, which is the 3rd sextile, and approximately the same as the 33rd and 66th percentiles, or the 333rd and 666th permille.
Income is not a normal distribution, so the two numbers don't tell much of a story by themselves. Usually, the statistics are shown as quintiles, plus median, 95th percentile, and maybe also 99th percentile and 999th permille, depending on whether the statistics presenter wants the audience to gasp or not.