I should have added, when you calculate the coordinate system, don't specify an output coordinate system. You want them to be in the shape's coordinate geometry as noted in the link
The output POINT_X and POINT_Y field values are based on the coordinate system of the dataset, not the coordinate system of the map display.
You don't want to do this part
To force the POINT_X and POINT_Y values to be in a coordinate system different than the input dataset, set the Output Coordinate System environment.
... sort of retired...