No, they are looking in the SAME direction. SOUTH! South is defined as the opposite direction to North. From where they stand, South is rimwards for both of them. The Southern Cross helps you find SOUTH.
The calculation helps you find a DIRECTION. Not a singular point. That DIRECTION is SOUTH. It is the SAME for both of them.
Imagine one person standing in Australia and looking south. At the same time another person is standing in Argentina looking south. Are they looking in the same direction? According to your map, no. They have their backs to each other. It is not the same for both of them, thanks to your silly map.
According to RET, it is likely day time in Argentina when the Southern Cross is visible in Australia.
I don't disagree with that, I was disagreeing with Thork's statement that on a flat earth map, south is the same direction for all observers.
Round earth - south is the same direction for all observers. North is the same direction for all observers.
Flat earth - south is a different direction for all observers at different longitudes. North is the same direction for all observers.