Pivot does not move when I change it to 'world' or 'parent'

I have a question about the objects pivot pivot, reading the manual and watching some random videos, I was under the impression that when I select a child object (object that is parented to another) and set its pivot to “parent” (tools options ‘axis orientation’), the child objects pivot will be moved to the exact position of the parents pivot, in effect using the parents pivot. Also setting it to “world” worked similarly, that is the child objects pivot is moved to the scenes origin point.

In my test scene that I have set up the above does not seem to be the case at all, you can see the parent object pivot is off to the left, when I set the child objects ‘axis orientation’ to ‘parent’ its pivot does not move at all, same thing when I change it to ‘world’, the pivot does not move at all. Have I misunderstood things here?