To be exempt from those requirements, a person has to have a medical condition that contraindicates vaccination.
What has caused some genuine confusion is that some states of Australia (not the federal government) do allow an exemption based on having had COVID in the last six months. This applies to moving around their state - but not to entry to the country (which state governments aren't meant to have any direct control over).
The federal government though has set entry requirements to Australia on the basis of the rules set by the most restrictive states, not the least restrictive, to avoid a situation where a person could legally enter the country in Melbourne, but couldn't do so in Perth (for example) - a single nationally consistent standard.
Under Australia's constitution the state governments have control over public health matters, so they set the rules. The federal government has to base its rules on the state rules for consistency. That's what has happened here.
A land of sheeple.