"Digital Exhaust" is being misused
From what's described, it sounds like they're misusing the phrase "digital exhaust." Twitter tweets are human expressions of information, opinion, and insight. It's Twitter's social network, and to my knowledge it has never treated tweets as "exhaust." That data is their life blood, and they've made APIs to that data widely available. Aggregators capture that data and offer it as well.
The term "digital exhaust" was coined to describe the data that most enterprises used to routinely throw away. It was described as exhaust because it was not seen as being "business data." Log files, for example, contain super high-scale machine data. IT might have kept this data on a short-term basis as a way to monitor the functioning of IT systems.
Now that Hadoop and other high-scale platforms are available, it's possible, economically speaking, to store this "exhaust data" and analytics firms such as Splunk and others have found ways to correlate this data with business events and customer data to develop business insights. To sum up, Tweets aren't exhaust data. That term describes types of data that used to thrown away because there was no perceived value and there was no economically sensible way to store that information.