Is it possible to load Twitter.app instead of opening Safari when clicking Twitter notifications in the Notification Center?
The current behaviour really annoys me because it keeps making me sign in and I don't want it to remember my credentials.
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Indeed, the Twitter notifications will always redirect you to the web application, even if you have a native Twitter client installed. In fact, Twitter notifications also work without Twitter.app as long as you have your Twitter account configured.
There currently is no way to change this behaviour.
If you use Choosy, you can redirect twitter.com links from anywhere to the official Twitter app (which Choosy recognizes as a browser). It won't open the specific tweet you've clicked on, but it will at least open the client.