The Jacobites were named after James VII of Scotland & II of England, whose first name in Latin is Jacobus. We see some iconography and inscriptions on coinage and artistic or monumental tributes referring to Jacobus or Iacobus, but I have never seen any evidence that contemporaries either in the Stuart court or outside of it would have called him anything other than James – unless they were writing in Latin – or Jacques due to the exiled court's intermittent relocation to and close ties with France.