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In the Screen Configuration settings, you can designate different Presenter screens (a.k.a. what Presenter projects) to various physical displays (i.e. projector, TV, external monitor, etc.) In the example image below, there are two external displays connected to Presenter, identified as Display 1 and Display 2.
To add a new screen, click + Add Screen. If you have a third display hooked up, you may designate the new screen to that Display 3 (not pictured). However, not all churches have such an option. That's where the Window option comes in handy, particularly if you want to capture that window for your online service. (You cannot set Main Audience Output or Stage Display screens to Window but any additional screens are eligible.)
In the image below, a new screen labeled "Livestream - Lower Third Lyrics" (henceforth referred to as "Livestream") was manually added. It's not pictured, but let's say Main Audience Output is already projecting to Display 1 and Stage Display is projecting to Display 2. The Livestream screen, then, has to be set to Window.
Doing so opens a pop-up window on the computer that is running Presenter. This essentially mimics having a physical third display. You can then set OBS (or the video controller of choice) to capture that specific window for the streamed service.
You can even change the format of the text being displayed on this window using the "Override foreground template" option under that screen's display settings. Per the settings in the example image above, Presenter is set to override the default template with the "Lower Thirds" template on the Livestream screen. You can see this in action in the image below - these projections are happening simultaneously, but notice how Presenter and the main Display Screen have centered lyrics, while the Livestream window's lyrics are in the lower third of the screen, mimicking subtitles on a movie.
To learn how to effectively use this separate, lower thirds template over a live recording, check out the article on livestreaming.