The push notifications are forwards of your onsite notifications (as would be the email notifications, though we don’t have them on this site).
Your examples already seem to be covered by the existing notification levels?
- Setting an individual topic to Tracking would give you counts in the topic lists of new posts for that topic, and wouldn’t generate notifications unless you were directly replied to/quoted, etc.
- While, as has been mentioned, Watching First Post at the category level would send a notification for newly created topics in that category (eg #testers).
- And Watching would generate a notification for all new replies in either a topic, if you set it at the topic level, or category if you set it from the category level.
There’s more info on all the different types of notifications in Notifications