How can I customize which Gmail notifications are shown in the notification bar?
You can filter messages based on sender, recipient, subject and other criteria from within the Web app. This is not possible from within the Android app. 


In a web browser, Goto [Settings] -> [Filters] of Gmail. Click on [Create new filter].
Create a filter to match your newsletters.





Assign a label to them (create one if you don't have one yet). Make sure to apply "skip the inbox (archive it)" such that this message only lands in that label, but doesn't land in your inbox triggering a notification.




Now go to label settings in the Android app and customize notifications for the "newsletters" label.