Every snake swallows it’s food whole and every snake (except teethless eggsnakes or wormsnakes or sum such freaky things) bite.
Constrictors (no venom, suffocating the prey) do bite and then wrap around the prey, that the snake tries to hold onto the frog seems to indicate it being a constricting species.
Venomous species bite and then often let go, as usually the venom is enough to kill or stun small prey and using your mouth to hold onto something that has the ability to fight back is usually a bad idea if you can just let the venom do it’s thing. Though many venomous snakes with less potent poison will also constrict their prey.
Most snakes, big and small are constrictors. Everything bigger than a king cobra is a constrictor, but most small species (like rat snakes, milk snakes, corn snakes and king snakes) are also constrictors.