The mini-Fiddy gets its name from the PAM127, the original Fiddy using a case design which first appeared in the 1950s, released in 2002. It came about since people loved the Fiddy's case shape (which copied from the 6152/1) but found the size too big. Hence necessity being the mother of invention, the 44mm mini-Fiddy was born. I believe the first mini-Fiddy's were produced by Getat, one of the pre-eminent homage watch makers. Also another feature of this watch was the use of manual-wind movements as well as a domed crystal.
The modern 1950s case used by models such as the PAM312 and PAM212 had a very similar shape to the mini-Fiddy, but this time used automatic movements, although the PAM212 and similar models also had a domed crystal with the PAM312 family having slightly curved crystals. There is also the vintage 1950s case
What is called the Luminor case (without the 1950) is more specifically the Bettarini case, designed by Alessandro Bettarini during Panerai's rebirth. It is the iconic Panerai case design and what differentiates it from the 1950s case are the sharp edges when viewed from the side, compared to the curved lines on the 1950s.
