Extremely well-written! I wished every mathematician would write as clearly as you. Thank you very, very, very much. :)

Thanks. Your definition of "homotopy from $H$ to $H'$" doesn't refer to $H$ and $H'$ again, which confuses me. What's the domain and what's the codomain of $K\colon X\times I\times I\to Y$ (considered as a homotopy between homotopies rather than a map between spaces)?

Thanks! I can't find a definition of "homotopy between homotopies" in the links you give. (At least I can't spot a sentence saying "If $H$ and $H'$ are homotopies from $f$ to $g$, then a homotopy from $H$ to $H'$ is a ...".)

Thanks, Zhen Lin!

Thanks! Can you prove the homotopy hypothesis with that definition?