I'm pretty sure Rachel doesn't get the cane. It's for punishing Amy. You should be able to give Rachel enough pain with other methods, and other elements of the story will help you progress.
Yeah I got the name wrong.. I am still a little confused, I seemed to get stuck in a state where I couldn't progress anyone. The second football skins vs shirts scene didn't trigger and I am not sure why. In the end I loaded an old save where I had already unlocked more. It might be as simple as upgrading the pool before the cafeteria. The hints get a little sketchy.