B767 Captain reported releasing the brakes without acknowledging receipt of the dangerous goods due to not receiving the form and summary.

Narrative

While in ZZZZ; we were finally cleared to depart after a several hour delay. Upon brake release and call for pushback we received the automated notification that we needed to acknowledge receipt of the dangerous goods pilot notification form and summary. We never received one. Since we were told to hold at the gate for 10 minutes by Ground; we had time to call Operations to find out if Dangerous Goods (DG) were loaded; but they never got back to us. We ACARS'd Dispatch and Load Planning as well to confirm whether or not we actually had DGs onboard. In the meantime; we were cleared to push and commenced push at the same time the DG pilot notification form and summary were printing. I sent the dangerous goods acknowledgement code. Everything happened very quickly and it was only later that I realized we were not to move the aircraft without sending dangerous goods acknowledgement code. I was also unaware; until preparing this report; that we actually have a dangerous goods acknowledgement code that is used to pull up the DG summary ourselves.