Yes that is Pss Diana saphire brooch. She did not like it very much as a brooch, so latter on she had it reset as the center for a small pearls several strings collier de chienne, and it became one of her most iconic pieces of jewelry. I had read many years ago that the saphire had fist belong to Queen Alexandra, and later to the Queen mother, who gave to Diana as a wedding gift. Since I have no reliable notes on this, I trust your notes that it was bought by the QM to Pss Diana. Any reference on where was it bought? Such a huge saphire would have been notest on catalogues or jewelers.
As to Joye's picture, I believe its the brooch in picture bellow from Getty archives
The picture is from april 1997, so it is impossible that is Pss Diana's saphire and QM has this brooch from many years, I've seen pictures of her in the 1960's wearing it. I believe it's an amethyst, but over those floral patterns it is not easy to tell for sure.