Elon Musk is a high-profile individual so his words can impact everything.
But NO, he's not manipulating the market because after all, why should he? His tweet is based on his view and people are riding it. If there's manipulation, he can move his coins but there's nothing such as that, and what he's just doing is making some shill tweets.
Investors are not dumb. Since Elon creates hype, they know for sure that there will be a time that he will also do another way around and some investors take advantage of that.
I share your opinion and I also think that manipulation requires some element of intention. For example if Musk makes a tweet or let's say a bad tweet knowing that Bitcoin will go down a little and before that he shorts Bitcoin with leverage, that is manipulation of a market to me because you take an action you seek to benefit from at the detriment of other market participants. I am only talking about markets here.
If he doesn't benefit from it at all because he doesn't have substantial positions, it is more like him having fun and laughing at himself wondering why his words can move trillion dollar markets.