No it’s not. The majority of power comes from the engine. It drives the electric motor mechanically, using it as a transmission. It is not just charging the battery.
There's a video linked elsewhere in this thread that explains how the toyota system works, it's worth watching as it clears up several common misconceptions about the system.
... and instead drive one of the motors, generating electricity to power the other motor to push the car forward.
I feel like that video you're mentioning did a poor job if it didn't describe one of the more important and clever aspects of the Prius's design.