Does the permissions popup for Push Notification in iOS refer to both local and remote (APNs) notifications?
Yes, preferences under Settings / Notifications work too for Local notifications. 

Apple's Documentation states that: 


  [The user] can also selectively enable or disable push notification types (that is, icon badging, alert messages, and sounds) for specific applications. 


Which is not exactly clarifying, but, I've just tested it with a small App which only has Local Notifications and, disabling all possibilities do disable Notifications. 

Bear in mind too, that an App which only has Local Notifications will be shown too among the Notification's Settings, which makes no sense if those settings won't affect that App.