The third scenario is the case where you hit the back button when you are the main screen of an app. Some app or games confirms the exit of the app with yes or no option..some do not have that..it will bring you back to the menu drawer or the home screen if you have launched it from the home screen or notification bar. So thats normal. If you are in the main activity screen of an app I would expect to show the home screen if i press the back button. Thats seems intuitive for me. And if back button exits an app without any confirmation you can still find it in the recents tab.
However in Lollipop you have Screen Pinning option available under Settings-> Security which allows you to stay in that particular app and back button will be disabled until you unpin the application. And you wont be able to come back to the home screen on pressing the back button from that application. But you can't switch from one application to another application without unpinning the current app.
If you feel you dont require the default back button actions anymore, Here is a related app in PlayStore. Back Button(No Root). You can turn off/disable the virtual or navigation bar back button completely if you want to using this app. The floating bar back button work fine in Lollipop but I was not able to enable the Navigation bar since it doesnot work on android versions 4.4.4 or later.
Screenshot (Click to enlarge)