The site's wording does not give the length of the rainbow progression. Any two integers form an arithmetic progression, so with progressions of two terms every coloring of $\{1,\ldots,k\}$ has a monochromatic $k$-term progression (one color) or a rainbow two-term one (two or more colors), while the one-color coloring of $\{1,\ldots,k-1\}$ has neither; read so, $H(k)=k$ and the displayed question has the answer no. A minimum of three terms would give yet another question, which no source states. The change inserts "$k$-term" before the rainbow progression; nothing else changes. The evidence is the poser's own text: Erdős and Graham (1979), printed p. 333 (Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics), define $H(n)$ by requiring that "there is always an $n$-term arithmetic progression all of whose terms either belong to one class or all different classes", and add that showing $H(n)^{1/n}/n\to\infty$ might be much harder; the site's wording renders that passage with $k$ for $n$. Bae (Section 2.1) and Fox and Hunter (Section 1.1) define $H(k)$ the same way. The omission is the site's. No result about the readings with a shorter rainbow progression is recorded.