I see. I must confess that I had never realized that a label which has an anchor akLeft but Alignment taRightJustify keeps its right edge fixed when the Caption is changed. Almost sounds like a bug, but I checked with Delphi, and it shows that same unlogical behaviour.
Anyway, if you want a label to be right-aligned and keep its right edge fixed then you should set its akRight anchor, as dsiders already said. And I would even go a step further an use the anchor editor to anchor the right edge of the label to a fixed point, e.g. the right edge of the parent.
I am attaching a modified version of your first project in which the left labels are anchored to the left side of the tabsheet, the center labels are anchored to the center of the tabsheet, and the right labels are anchored to the right sides of the tabsheet. I also painted the background in yellow to exactly see the position of the labels.