Have I committed a sin by purchasing a train ticket at a reduced price, knowing that the ticket machine was out of order and I had to purchase the ticket inside the train, where I obtained a ticket at a lower price than usual?

If the ticket allocated to another category is cheaper than the regular ticket, the conductor is not allowed to give it to you unless he is authorized to do so. If you know that he is not authorized, then you must pay the difference to the authority responsible for the train.