Peterson has a line drawing of this badge on page 93 of his book. Yet he doesn't
mention how exceedingly rare this badge is. One source I found claimed that only 21
were ever awarded, which would make this the scarcest IJN badge without a doubt.

The second version of this badge was instituted in 1911 and has a different design.
It is below.

The Model 1891 badge was instituted on June 2, 1891 and abolished (replaced by
the later badge) on September 30, 1911.
Navy Staff College Graduation Badges
Models 1891 & 1911
海軍大学校卒業記章
Here is the 1911 model badge. Thanks for the pictures, Brenton!

The Model 1911 badge was instituted on September 30, 1911 and discontinued on October 23, 1922.
I don't think there was another badge to take its place, nor am I aware of why these badges were discontinued. I read in
one Japanese source that only 234 of these badges were ever awarded.
Model 1911 case