Cartographica provides a number of different ways to show coordinates for features. In the case of points, there are three primary ones, each with their own advantages.
The Point Data window is accessible from the Windows menu and shows all parts and points for the selected geometry. It is useful for all feature types (points, polygons, and lines) and shows the coordinates in the coordinate system of the layer itself, as opposed to the coordinate system of the map being viewed.
The Point Data window is a read/write window, so you can add/delete/edit points by modifying the coordinates in this window. Changes take effect immediately and are undoable.
Add Coordinates Columns adds 2 columns, one each for X and Y coordinates of the features in the layer. This command adds this data to the layer and will be saved as a modification when the map set is closed (assuming you don't cancel the save). When used, the Add Coordinates Columns command adds two new columns and sets the column data based on the current Map coordinate system. As such, it can be an easy way to add coordinate columns for multiple coordinate systems. For example, if you have feature data in UTM and want to have both UTM and Latitude/Longitude coordinates in your data area, you can do so through the following steps:
The last method is a blend of the two previous methods. It provides dynamic columns that display data based on the current Map CRS which change when the CRS is changed or when the data points themselves change. It's a bit trickier to set up.
Bonus hint: if you use geometry.centroid(1).x and geometry.centroid(1).y the original coordinates of the layer will be used instead of the projected coordinates.