This is correct. However, at the time Bella was not aware of this. Only later when asked by the Volturi did Bella realize that Renesmee was not venomous. Bella was dying from giving birth to Renesmee because her body could not take the trauma of the baby being ripped from her body. This is why Edward stood by ready to inject Bella's heart with his own venom and why immediately bit her in as many places as he could, to keep her from dying.