Ah, no, this is some Internet slang, and oddly enough it comes from the first meaning. AFAIK, the second one doesn’t exist in Japanese.
Basically, “hahaha” in Katakana is written as ハハハ. If you line up enough ハハ’s, it will look like a series of ｗ’s. In chats, they use ｗ (from 笑い、warai) to denote laughter. If you line up enough ｗｗｗｗ’s, it looks like grass. That’s how 草 ended up meaning LOL.
