That’s all you’ll need (the 2 initial vax). The process is quick, just a glance at those documents. My international vaccine certificate (Australian one) has a QR code on it but they don’t scan it - just glance that you brought it along. On a recent trip they didn’t ask for photos of our negative tests, but I had them just in case. Boarded in Sydney, whereas Melbourne and other Aussie ports will likely be more thorough.