That's the problem though, They are saying that the email sent by my girlfriend effectively says "we want to move out when the contract ends in October" which to you and me might not but to them it's clear as day (or a way they can make more money with BS loopholes)
There is no loophole. You did not give them formal notice in writing. They can claim to have interpreted her email in anyway they like but the reality is, you did not give your notice in.