reset or delete app across devices

I don't know how best to explain the problem I'm having.

Background:

I have an iPhone 5 with my own apple ID, and I have an iPad mini for my kids with a separate apple ID. When I first set up the iPad mini, I had set it up with my ID, but soon realised my mistake and created the ID for the children and got all apps on their ID. Also have an iPod 4 that uses the same ID as the iPad, and have apps for that ID set to sync automatically. None of our devices back up to the cloud, only locally.

I installed the game Despicable Me: Minion Rush for the kids (with their ID) and on my iPhone. But after a while I realised that on both devices, it seemed to be picking up the same game -- the kids (who are too young to read) were just clicking buttons, and must have been asked if they wanted to use the game from the cloud vs local, and chose cloud. I only realised this when I saw the same question on my phone, and chose local -- although didn't understand how it was in the cloud since I have cloud settings off for my phone.

The questions:

I want to be able to play my own game, without the kids accessing it from a separate device. I've tried logging out of Game Center on all three devices. I've deleted the game from all three devices AND from iTunes on my PC (all three devices only back up to my PC), but when I go into the AppStore, I still get the cloud / download icon rather than the option to download it Free (meaning it is a new app). When I download it, it immediately restores to my previous game scores ("We detect your remote saved data").

How do I 1) completely remove any history of this game from my IDs so we can start fresh with it? and

2) ensure that the devices remain completely separate (when I have no idea how they got connected in the first place!)?

Thanks in advance!