How to prevent an app's notification from updating while the screen is off?
The preferred way would be if your media player has an option in it's settings to disable notifications, do it that way. 

The other method would be to use the embedded features of Android to disable the notifications, when a notification from your media player pops up, pull down the notification shade and long-press the notification tile, and select "Stop Notifications". This will disable the notifications by taking away the apps permissions to do so. To re-enable notifications, go to Settings - Apps and find the apps, and turn the notifications back on there.

Otherwise the device is working exactly as designed with the settings you have.