Hi Gianni,
All tracker-related methods can only be used after initialization, meaning you have to call SetGeometryWildcardValues after Initialize(). I know this is self-evident, but just remember the rule of thumb: trackers are used to modify the states (pos, vel...) of the simulation objects, and when you need to modify the states of objects, the simulation world has to be "initialized" first so these quantities exist; while simulation object definitions, such as InformCentroidPrincipal or material property assignments, are in general done before initialization.
As for InformCentroidPrincipal, I realized that the comments for the mesh version of the method have some copy-paste artifacts which I'll fix soon. But I think your understanding is basically correct (with a loose usage of "geometrical center"). I'll write down how I see these methods should be used, and I think your notes regarding the principal axes are equivalent:
These two methods are used w.r.t. the underlying frame you created the mesh with. If you want fewer brain teasers, it's preferable to choose to create the mesh with the CoM of the object that the mesh represents at (0, 0, 0) and its principal frame aligned with XYZ. If you do this, then you never need to call InformCentroidPrincipal or Move. However, if you did not do this during mesh creation, then InformCentroidPrincipal or Move can be used as remedial measures. You can use InformCentroidPrincipal to notify the solver where the CoM and principal frame are in the mesh's own underlying frame (then the mesh will be modified accordingly under the hood). I wouldn't suggest using Move as its name does suggest something ambiguous but it's intended for the same purpose: using it here means rotating and then translating the mesh in the hope that after this, its principal frame agrees with its underlying frame. Whether you call the methods or not, after the simulation starts, the meshed objects' simulation world pos/rot always means the pos/rot of its principal frame.
And it's correct that you can use SetMOI to fully define the object.
Thank you,
Ruochun