One reason for the Aus/US difference is that in Aus there is only one definition of MSA for the route sector and that is what it says it is. The sector has an MSA published and that altitude is good for the route between the fixes defining the sector with about a 4nm buffer each side and at each end of the sector. So you could theoretically cross the fix into the next sector with a higher MSA and then climb, provided you could reach the new MSA within the 4 mile buffer. In practice however, nobody would even try and so the rule is that you should achieve the new MSA before crossing the fix.
Same thing applies when going from a higher to a lower sector MSA. Don't descend until past the fix.
In the US it is the same if the sector is based on MSA, but often it is based on MEA or Navaid Reception and not MSA; in that case it is not imperative to climb beforehand. If it is based on MSA/MCA then it is stated so on the chart amd an early climb would be required.
If a US pilot is flying anywhere else in the world, he should bone up on this.