From what I understand, Twitter only posts content from the linked site (expanded tweets) if the site is a partner or (twitter cards) if the site actively includes code for Twitter. Anything you can squeeze into 140 characters is surely protected as a (very) limited quote.

Which makes me question this article. Did they really pass a law requiring payment for quoting small parts of an article? Seems unlikely. Paying for merely linking to a site would be even crazier.