Based on my own experience and eclectic education, "ku" means "verse" or "stanza". I've heard of "hai" being more "play" than "amusement", though the difference is somewhat ephemeral there and depends greatly upon context.

Oddly enough, in application "haiku" has come to be less about amusement than about kensho (a zen buddhist term relating to an epiphanic moment of self-knowledge, similar to the more general enlightenment of satori). It has become a traditionally zen meditation in poetic form. Haiku typically involve imagery associated with nature.

The senryu form has the same syllabic stanza layout as the haiku, but tends more to have its focus directed at human nature than nature in the more general sense, and senryu are often more "amusing" than haiku as well (even outright funny at times).

Yes, I have spent entirely too much time thinking about stuff like this, and reading about it. The haiku is actually my favorite poetic form.