With regards to the Maximum Power Theorem (Princeton.edu);
The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero.
I don't know if I am making the same "mistake" that Joule made, however I still don't see how at maximum power transfer, the efficiency is anything but 50%. At maximum power transfer, the source impedance = load impedance, hence equal power dissipated in the source and the load. Am I missing something here?