They use the older "Silver-C" and "Diamond-C"
nomenclature. What did the C stand for?

This originates from the stone age of soaring on the Wasserkuppe in the
1920ies. Fritz Stamer developed a pilot training syllabus which
contained an A, B and C test and the respective badges (one, two and
three seagulls) which the pilots wore with pride. The respective
requirements for those tests changed over time, in the beginning, the A
test was something like a 10 seconds flight or some such. In the
beginning the C test was the pilot's licence, later an official licence
was added. BTW, in Germany the A, B and C tests are still in use, the C
sometimes also called the "bronze C".

As soaring evolved, the "Silver C" badge was introduced. Wolf Hirth was
the first to receive that badge in 1931.

Even later Gold C and the diamonds were added to keep up with the progress.