The site's parenthetical gloss says only that $\chi(G)\leq3$; it does not express the preceding exact condition $\chi(G)=3$. Taken literally, the gloss makes both questions false: arbitrarily large stars are intersecting and $2$-colorable, and every two of their edges meet in exactly one vertex. The site's commentary on Alon's counterexample states that "its chromatic number is 3", and Erdős and Lovász [ErLo75, Theorem 8 p. 613, construction (b) p. 620] work with exact chromatic number $3$ and count only points lying in edges. The corrected Statement adds the missing "no such $2$-colouring" and the convention that every vertex lies in an edge; if isolated vertices were allowed, adjoining isolates would trivially falsify the first question even under the exact chromatic-number hypothesis.