Is it permissible for a teacher to stipulate to the client the payment for sessions canceled by the client, or to contract with him on a monthly basis as schools do, or to consider the time allocated to the client as a down payment for missing the opportunity to sell this time to another client?

There is no blame upon you for demanding payment for the lessons that the student or their guardian apologized for, as long as the excuse came from them and not from you. If the contract was on a monthly basis, then you are entitled to the payment upon the passage of time, because you are considered a private employee, and you are entitled to payment by enablement, whether work was done or not. However, if the contract with you was per lesson, then you are not entitled to compensation for the lessons that were not taught. An earnest payment (down payment) is permissible in leasing (ijarah (definition)), just as it is valid in sale.