The site's $R_n$ is one number for each $n$: the largest radius whose disc almost every walk covers by time $n$. The change replaces "almost every random walk" by "the random walk", so that $R_n$ is the radius covered by each path, and inserts "in probability": for every $\varepsilon>0$ there are constants $0<a\le b$ with $\mathbb P\bigl(a\sqrt{\log n}\le\log R_n\le b\sqrt{\log n}\bigr)>1-\varepsilon$ for all large $n$. The evidence is the poser's own statement of the question, Some of Paul's favorite problems (1999), Problem 6.76 (Erdős, Taylor), printed p. 12, on the source page: it lets $\xi(x,n)$ count the visits of the walk to $x$ up to time $n$ (printed p. 11) and takes $R_n$ to be the largest integer with $\xi(x,n)>0$ for each $\|x\|\le R_n$, a quantity of the path with no "almost every". The same item conjectures that "$R_n$ is about $\exp((\log n)^{1/2})$" and calls Kesten's limit law $\mathbb P\{(\log R_n)^2/\log n<x\}\to1-e^{-\lambda x}$ the "stronger conjecture"; the site's commentary also calls that law the stronger conjecture. A limit law with that continuous limit implies the two-sided comparison in probability, so the comparison in probability is the reading of $\asymp$ that the law strengthens. Dembo, Peres and Rosen (2007), equation (1.1), reprint p. 1, define $R_n$ in the same pathwise way. The defect is the site's: the 1999 statement has no "almost every".