The “Add to home screen” option on suppoorted browsers installs the Progress Web App (PWA) of the forum. Essentially it’s a small app that runs on the existing browser, not just a shortcut.
Seen here, my PWA was created by Chrome and it uses the Chrome (Chromium) name in package.
I also receives live notification in the PWA, just need to toggle it in Preference/Notification.
I don’t understand the notion behind using the official “full” Discourse app. It’s buggy, hence the 2-star review on both Android and iOS. It’s also large and clunky (install size: 30 MB full app VS 0.4 MB PWA, full app is nearly 100-times larger than PWA). Besides, the full app hasn’t been updated in half-a-year, while PWA can be updated independently whenever you login to sync up with the newest upstream code.