There's something screwy about those Stellarium pics. They can't all be photos, because it's not possible for it to be night-time in all four locations simultaneously. In particular, the sun doesn't set on Argentina at four in the afternoon!
So I think some (or all) of them are just artist's impressions of where the stars would be if you could see them at that moment.
As for the displacement of the Southern Cross constellation, what the map at the top of this post doesn't illustrate very well is the latitudes of the four locations, relative to each other.
Christiana (South Africa) is 27.88 degrees south of the equator.
San Juan (Albardon, Argentina) is 31.25 degrees south of the equator.
Kalgoorlie (Australia) is 35.74 degrees south of the equator.
Auckland (New Zealand) is 36.84 degrees south of the equator.
So Australia and New Zealand are the most southern locations, and fairly close to each other, and that's why the Southern Cross is in roughly the same position when viewed from either location.
The next "most southern" location is San Juan, Argentina, which is five degrees further back toward the equator. That's why the Southern Cross is clearly lower in the sky when viewed from that city (in fact, the horizon is effectively higher, relative to the constellation).
And finally, Christiana (South Africa) is the most northerly (closest to the equator) of the four locations, and so the constellation is blocked by the horizon.
The difference between the degrees of latitude for each city sounds tiny, but each degree of latitude works out as 111km on the Earth's surface, so Christiana is 994.56km "higher" (further northward) than Auckland.
That doesn't sound like enough to make a significant difference, but it's nearly 1,000km, and the entire distance from the North Pole to the South Pole is only 12,000km. I.e., it's about eight percent of the maximum possible distance.
(I had no idea this reply would end up being so lengthy when I started typing it, so apologies for that.)
The locations are taken from the southern hemispheres. Their exact degree below the equator is irrelevant because the point is not where at in the sky the constellations rest relevant to each other; the point is that you can see the same constellations at all. I would love for you to check out the accuracy of the program on your own computer. My cpu was acting funny as I was looking at the pictures. It took a long time to finally get them right, but I'm certain they are accurate and represent the skies from the different locations at the same julian date and roughly the exact same zulu time, give or take a couple minutes.
Well, as the examples you gave show, if you're in Christiana (South Africa) you can't see the Southern Cross at the same moment that someone in Auckland (New Zealand) can see it. So each city's exact degree of latitude below the equator clearly is relevant!
I get the feeling we're at cross-purposes here, so will drop the subject and wish you well with your inquiries.
I will use a different constellation to make the original point. Look at the lyre on the butt of the centaur. it is visible from each direction. If people hold hands in a circle and all face away from each other, they shouldn't see the same thing.