Technically, the single tweet did not send the shares soaring. A significant investment from successful active investor Icahn did. The tweet just revealed the investment to public.
Take Elon Musk's tweet that Tesla was being profitable for the first time as an example. The tweet lead to soaring stock price but the cause was excellent execution by Tesla team, not the tweet itself.
In this case you're probably right, but lots of HFT algorithms do use Twitter as a feed to inform trading strategies. Tweets can literally send stocks soaring or sinking.
I think it's a combination of both. Icahn's investment in Apple was the reason that people found reason to invest, but that information was brought to people through the distribution channel of Twitter.
So Twitter is an accomplice in this situation, since it acts as how that information got disseminated to an audience, which I think still counts for a lot since it could have been an important signal, but it without Twitter, not as many people might have not known about it.
Not only that, but the fast dissemination of this information (due to the nature of Twitter) might have made the stock go up faster than a news article coming out a day later, being read at different times of the day.
Take Elon Musk's tweet that Tesla was being profitable for the first time as an example. The tweet lead to soaring stock price but the cause was excellent execution by Tesla team, not the tweet itself.