Compare figures 1 and 2. They depict two triangles such that the shortest distance between them is the same. However, qualitatively, the two triangles in figure 2 appear closer than the two triangles in figure 1. This is what the website calls the insensitivity of the shortest distance metric to position (the way the shapes are posed in the ambient space).
Now look at figures 4 and 5. Figure 4 depicts the same triangles as in figure 1, and figure 5 depicts the same triangles as in figure 2. Now, what is computed is the Hausdorff distance instead of the shortest distance. It is now seen that the Hausdorff distance between the triangles in figure 4 is larger than for those in figure 5. Thus, it is seen that the Hausdorff distance captures the initial intuition that the triangles in figure 5 are indeed closer.
This property, that the Hausdorff distance considers the position (again, the way the shapes are posed in the ambience), is what the website calls the sensitivity of the Hausdorff distance to position.
I'm not sure this terminology is very illuminating. I would say something along the lines of "the Hausdorff distance is sensitive to the global positioning of the two sets, and not just to the local distances".
A more straightforward example would be in $\mathbb R$. Consider $A=[0,1]$ and $B=[2,3]$. The shortest distance metric between $A$ and $B$ is $1$. The Hausdorff distance between $A$ and $B$ is 2. Now, consider $B'=[2,100]$. The shortest distance between $A$ and $B'$ did not change, it's still $1$. But the Hausdorff distance is between $A$ and $B'$ is now $99$. The shortest distance metric is blind to the global relative positioning of the two shapes, but the Hausdorff distance picks it up.