When is a contract cancelled?

If you have a contract for services that has a few months left to run on it, that contract would ordinarily not be cancellable early unless 1) there was some term or condition in it allowing early cancellation; 2) the company violated the contract in some material (important) way; or 3) you asked to cancel and the company agreed to let you cancel--but since that agreement is voluntary, they could impose conditions on it, and if you accept  those conditions, you'd have to abide by it.

Under the situation you describe, you did not accept their early cancellation conditions, since you did not provide an early cancellation letter or pay the outstanding balance at the time. Therefore, it would seem that your contract was not cancelled. You would be obligated for the remaining term of the contract--including paying for any remaining months--but would not have to make a lump sum payment of the balance; they will also have to continue providing the service while you are paying, for the remainder of the term.