B737-800 Captain reported an uncommanded aircraft movement occurred when the towbar broke during the pushback. The Ramp Supervisor decided headsets were not going to be used for pushback due to the lighting at the airport; which led to a communication breakdown between the flight crew; the Ramp Supervisor; and Ramp Control regarding the aircraft push direction. After the aircraft was pushed in the wrong direction the towbar broke; which led to the aircraft rolling forward uncommanded that was then stopped by the Captain applying the brakes.

Narrative

After the jet bridge was disconnected; the Ramp Supervisor indicated that; due to the lightning in the vicinity of the airport; they would not use headsets for the pushback. From the opened flight deck side window; I communicated to him that the Ramp Controller cleared us to push onto [Taxiway] XX with the tail east. He seemed confused and replied that normally they pushed either tail north or south; then my First Officer verified with Ramp Control and I confirmed to him that they wanted us with tail east; which he acknowledged.I gave him the brake released hand sign and he said clear to start engine number 2. They started pushing us; but they stopped with the tail south. Since the Supervisor was still outside my open window; I restated that the tail had to be east. He seemed confused. Therefore; I told him to wait while we checked with Ramp Control if they could leave us with the tail south. Shortly thereafter I noticed the aircraft was rolling forward and I had to apply the brakes to stop it. When I queried him as to what happened; he told me that the towbar broke and he repeatedly assured me that there was no aircraft damage and that he could remove the bypass pin. At this point; I don't know if the pushback was concluded prematurely because they were confused with the tail east instructions or because the towbar broke. Clearly there was a communication issue due to the lack of headsets and between the Supervisor and the rest of the ground crew who never asked me to set the parking brake and to whom I never gave the hand signal that the parking brake had been set.