It appears they are different tools. The Add Geometry Attributes will create new fields and calculates the geometry. The Calculate Geometry Attributes appears to update existing fields which is what I need.
I wonder if you are right, though, in the fact that the help document shouldn't exist for Desktop? It's interesting that they have the drop down at the top of the help page where you can change to whichever version of desktop you are using to get the applicable help document. In the case of the Calculate Geometry, it shows that versions older than 10.6 are greyed out, but 10.6 is a validate selection. The Add geometry help page shows that it's been around at least since 10.3.
Hopefully someone from @ESRI will chime in and clarify this.
Thanks for the help Joshua.