How can I turn on opening a link in another app again?
If I'm reading the documentation correctly, when the web interface pops up in Safari, there should be a banner across the top of the page that essentially says, "Hey, you could view this info in the app instead!" If you tap OPEN on that banner, it's supposed to re-establish the universal linking to the app instead of the website. Here it is in Apple's words:


  When a user taps a universal link that [an app] handle[s], iOS also examines the user’s recent choices to determine whether to open [the] app or [the] website. For example, a user who has tapped a universal link to open [the] app can later choose to open [the] website in Safari by tapping a breadcrumb button in the status bar. After the user makes this choice, iOS continues to open [the] website in Safari until the user chooses to open [the] app by tapping OPEN in the Smart App Banner on the webpage.


Source: Universal Links in Apple's App Search Programming Guide