I and others have tried to explain it several times, and you don't get it.
NOTIFICATIONS is the capability for an app to be awakened when its associated server has content for it. The server (facebook.com in this case) sends a message to Apple, and Apple forwards that message to the phone. The phone wakes the app and delivers the notification. The app receives the notification and does SOMETHING with it. What it does has nothing to do with iOS or the iPhone, it is entirely up to the app as to what to do. It may retrieve the content that is available (or it may not), it may display a badge (or it may not).
If you turn the main NOTIFICATIONS button off the app will not longer receive NOTIFICATIONS. But that does not stop the app from displaying a badge, because badges are under the control of the app. And, if the app has a background process (and Facebook does), it can still receive content even if NOTIFICATIONS are off. It can also retrieve content and display a badge when you launch it. Just like the Mail app does even if you have Push turned off.
If you go to Notifications > Facebook > BADGES > OFF.that tells the Facebook app that you don't want to see a badge. If the Facebook app still displays a badge it is not obeying your directive. That is the fault of the app developer. iOS HAS NO CONTROL OVER BADGES, The APPLICATION controls badges. Why can you not grasp this?
This will be my last response in this thread. I've said all there is to say, and more than once. I'm sure I'm boring other readers.