I have the same problem. SMS notifications show the phone number and not the persons contact name. It then shows the contact name when you open the text.I receive more text. Seem iMessage will appear as name in notification centre but SMS still appear phone number. Very annoy.
This sounds like a basic contact sync issue, not with iOS itself. It is working perfectly fine assuming it is set up correctly. Have you checked to see if you have contacts in the Contacts app and/or if iCloud contact sync is turned on?Hope it fix itself soon cos I have keep open iMessage thread see who it from. I usually quick look notifications if from main person. Can't do that right now.
Hope it fix without need wait iOS 9.1 or something.