On Windows Phone, the notifications aren't a two way sync between the phone and device. What happens is that when a notification comes in, the OS hands off the notification to the Health app and that's it. The Health app can't "see" directly what's in the apps, Action Center, or anything else that the OS didn't hand off to it.
After the Health app receives a notification from the OS, it then checks the notification against any filters that you have set (i.e. if it belongs to a certain enabled tile on the Band, email VIP, etc). In theory, if it fails to be caught in any filters, then it should be categorized as one for the Notification Center tile on the Band. Finally once as it knows where it should go on your Band, it attempts to contact it and relay the notification.
If the app can't reach your Band because it's out of range, usually it should queue up the notifications until it finally does reconnect. You can see this happening if you get a lot of notifications overnight and then exit sleep mode on the Band.
What puzzles me is why emails are working fine though but not others. My guess is that there seems to be some sort of problem with the Health app receiving the notifications properly from the phone. I'd say try toggling the Health app off and back on in the Accessory Apps section of the phone settings. Rebooting both the phone and Band doesn't hurt as well. If that doesn't work, then perhaps uninstalling and reinstalling the Health app might help.