With Precious Scars equipped and the Solar subclass, restoration *should* be given by getting kills with Solar weapons. This season, Solar weapon kills don't seem to count but Kinetic kills do.
The behavior seems to be correct on Prismatic with either Strand super + Strand weapon or Void super + Void weapon. Haven't tried anything else.
EDIT: The behavior seems correct on Solar if the Sol Invictus aspect is not equipped. Once it gets equipped, kinetic procs restoration instead of Solar.