"Pure Racing Edition" removes both coins and items (such as shells or mushrooms) from the GP and Match Race modes. All "?" blocks and coin pick-ups have been removed from the race tracks and the CPU controlled karts will not be able to use weapons either. You will not lose a coin from bumping other racers, Lakitu will not charge 2 coins to bring you back, etc.

"No Coins" ONLY removes coins from these modes. All item boxes are still in the game however you will never get a coin from an item box. CPU controlled karts will still use their weapons.

"No Items" ONLY removes item boxes from the game but leaves in the coin pick-ups. CPU controlled karts will not be able to use weapons. You will lose a coin from bumping other racers, Lakitu will charge to bring you back, etc. 

The Battle Mode and Time Trial modes are not changed in any version.