The site's wording (page last edited 2 October 2025) fails at $k=1$, which its range admits for every $n\ge1$: every line through one of the points is $1$-rich, so there are infinitely many $1$-rich lines and no bound $\ll n^2$ holds. The failure is this page's own elementary check, and it is the only one: at $k=2$ each $2$-rich line is determined by two of the points, so there are at most $\binom n2<4\,n^2/2^3$ of them. The change inserts "$2\leq$" before "$k\leq n^{1/2}$"; the upper end is Erdős's print and stays. The defect is already in the poser's text: Erdős [Er87b, Section 2, p. 169] states the conjecture of Croft, Purdy and Erdős "for $k\leq n^{1/2}$", with no lower end, and in the next sentence reports it "proved by Szemerédi and Trotter". Szemerédi and Trotter state their Theorem 2 for $k\leq\sqrt n$ (p. 382) and restate and prove it for $2\leq k\leq\sqrt n$ (p. 389), the form inserted here. That range is used because two sources corroborate it as the problem's form: Erdős's own report that the theorem settles the conjecture, and the site's curator, whose label SOLVED and commentary ("This is true, and was proved by Szemerédi and Trotter [SzTr83]") read the statement as the theorem proves it. The same range is also exactly the exclusion of the one value, $k=1$, at which no finite count is possible. The form was fixed from Erdős's report, the site's reading and the exclusion of $k=1$, not from the theorem's hypothesis range alone. No result concerns the site's wording alone.