Why does the Notes app show on the lock screen?
Handoff is why the notes app shows up on my lock screen.

You could try disabling that entirely or be sure that no other device is communicating with your device over bluetooth or WiFi.

If you look at the text in the handoff settings, suggested apps can also be shown based on usage patterns. If you open notes every night at 7 pm, eventually the OS will suggest notes at that time based on inference. Since this intelligence was introduced with iOS nine, it would take a while for the OS to learn your habits so having it appear coincidental  with the 9.0.2 update seems plausible.