Okay, so how do I go about determining which this is, and turn it back off from there? In any event, I have gone to Start > System > Settings > Apps, scrolled down to Twitter on that list, and selected Advanced options there, and then selected Set default apps, and then Choose default apps by protocol, for instance, selected Twitter from there, and the only options it gives is to select the existing Twitter app, or Look for an app from the store. There isn't anything resembling a means of dropping the app that is shown back out of consideration and reverting things back to having links go to the browser.
I DO notice that a step or two back from that, under the settings from Twitter there, directly above the Defaults header (where I can select to go to the Set default apps pane), that there's an option under the App permissions header to turn Background apps Off. Would that fix it?
This is also the pane where I'd reset the app back to default before, after which Twitter links went back to going to the browser again like they had for years. Nevertheless, the Twitter app quietly stole the thing back again a couple days later. Bah.