I agree with all the others. You were completely right and they don't have a leg to stand on. Just trying to force you to pay some money.
A few weeks ago, my father wanted to cancel our cable TV service, because we don't use it anymore. When he called (or went there? Not sure) to ask them to cancel our subscription, they offered to lower the price. He cancelled nonetheless, and then came back to tell us that even if he didn't have the intent to cancel before, he would have cancelled right after they offered to lower the price. He found it very cheap on their behalf that they'll keep charging you regular price if you're a regular customer of years, but they'll lower it if you just say you don't want the service anymore.
ETA: I was writing as you posted your update! Glad to know it's all solved.
