Hi John,
You are correct, «Rank» means the position of the story in the default sorting order. This order represents the priorities according to the Product Owner.
Actually, the rank is implemented as a property of individual stories that you can find in exports, API etc. However, what really matters in the user interface is the resulting list. The rank of an individual story has little value on their own. That is why it is not displayed in the story details panel.
Changing the rank on an individual story without the context of the whole list would also have little value, as a Product Owner usually needs to position it relative to other stories. That is why the only way you can change it from a story details is to use the «Move to top priority» menu, which does not require additional context. Otherwise, you can change it by reordering stories by drag and drop in the list itself (only the Product Owner can do that).
It is offered as an explicit sorting criteria so that the default sorting criteria is shown and so that you can go back to it if you changed it temporarily. It is not called «Position» because when the sorting order is different, e.g. «Business Value», then the actual visual position of the story changes and it not equal to its «Rank» anymore.
Does it make sense?