B737 Captain reported they began taxiing to exit the de-icing pad before being cleared to do so. The de-ice trucks had to move out of the way to avoid a possible collision.

Narrative

We went to the central de-ice facility prior to departure. Everything was uneventful up until we received our post application debrief. After receiving the debrief we were told to contact 'outbound' [Ground Control] for clearance to exit the pad. We contacted the 'outbound' controller and received clearance and taxi instructions. We did receive taxi clearance and began to move. After about ten feet of movement 'outbound' told us to stop and go back to 'iceman.' After stopping and contacting 'iceman' they informed us that we left without clearance and there were still trucks working at the back of the aircraft that had to move out of the way when we moved. Once cleared again to depart I asked for a phone number so I could talk to them on a land-line to figure out what happened. The 'outbound' controller kept ignoring my request and eventually just said 'bye.' All I can think of is that we responded/took someone else's post application debrief (although none of the other aircraft being de-iced spoke up at the time). Another complicating factor was that the electronic boards were not working properly.