Air carrier Captain reported brake release before acknowledgement of the Final Dangerous Goods Summary was sent.

Narrative

We were finished with the Before Pushback Checklist and had received the Final Dangerous Goods Summary. We were communicating with the ground push crew and trying to determine who we talked to for pushback clearance from with ATC. The ground crew were asking for brake release and I told them we couldn't release the brakes yet. I asked the flying First Officer if we acknowledged the Dangerous Goods Summary and he said that we did. He may have thought that I meant the Final Dangerous Goods Summary. We received pushback clearance from ATC; asked the ground crew if we could still release brakes; and were told that we could release brakes. I released the brakes and we immediately received an ACARS message to acknowledge the Dangerous Goods Summary which we then complied with before aircraft movement. We continued the pushback and flight without any further issue. The Final Dangerous Goods Summary was received around 4 minutes before brake release. The communication confusion and language difference in ZZZZ with ATC along with the ground crew insisting on me releasing the brakes were definitely contributing factors distracting me from sending the acknowledgement code so as to receive the DG (Dangerous Goods) Summary acknowledgement.