BTW, why do I have a browser window? Is it because Iam using Android, or I am running nodebb 1.14?
It's because you're running 1.14 actually. The actual scripts for installable PWA only appeared in 1.15.
However, browsers are confusing so they also offer the option to "add to home screen" any website, but instead of actually considering its manifest for how it's supposed to run and using the service worker properly it will be just an icon to open that website in the browser.
I would've recommended checking this website to see how it's supposed to work, but it seems to be broken in 1.16.2 unfortunately (mentioned in this issue - service worker was moved to its own directory so only paths starting with /assets/src/ are in its scope.)
EDIT: while it was mentioned in the issue #9193, it wasn't the main focus. I commented on it on #9127, solution to which actually broke the service worker
Otherwise I think it could be done via the hook filter:topics.addPostData? Just iterate over the posts array, get the category, and for every user of a post add an object to custom_profile_info with the property content set to some HTML indicating moderator rights?
Probably this... alternatively, use the groups method, but hook into filter:topics.addPostData to remove the group badge if they're not in the category that user moderates?