For best current overview look at the IATA site - https://www.iatatravelcentre.com/world.php as well as the Australian government sites .. I wouldn't be relying on BA ( or any other airline ) to keep their own site up to date I'm afraid

If you are vaccinated and can prove it then Australia is easy now (finally!) .. bring the Vax cert and 'sign' the digital form and that's basically it ( check the individual states too as there may be additional forms, but no additional rules any more I don't believe )

On the 'eVisitor Visa' vs. 'Electronic Travel Authorisation' .. far as I know you need one or the other not both- they are functionally the same ( it depends on your passport which ones you can apply for , some nationalities can apply for both )