In my experience, Contents and Primary Text are the same....whatever is in Primary Text shows up when Contents is displayed.  Name is a separate field which shows when viewing artifact links.  We have found that the Name and Primary Text fields can be synchronized or un-synchronized.  If synchronized, whenever you change the Primary Text (or Contents) fields, the name also changes.  You can tell that they are synchronized because the content in the Name field will have < > around it.  However, if they are un-synchronized, then any changes you make to Primary contents will NOT be reflected in the name field.  If you use artifacts for requirements like we do, this is undesirable, as I want to see the requirement accurately when viewing links.  To re-sync the two fields, you have to edit the artifact by opening it up, deleting the content in the Name field and then saving the artifact.  It will automatically re-populate the Name field with whatever is in the Primary Content field with the < > around it.  Another way to do it is CSV import/export (assuming you are on DNG 6.0.2 or better which supports round trip import/export). Export your artifacts in CSV with the Name and Primary Text columns showing.  Clear all the name cells and re-import the CSV , updating the artifacts.  viola!

 I probably have the reverse problem, but perhaps the same solution.