Clustering coefficient (equation) for a regular ring lattice

Hi, I would like to understand why the clustering coefficient for a regular ring lattice has the following equation: $$C(v) = \frac{3(d-2)}{4(d-1)}$$. Do you know how to derive it? Or where I can find a derivation of it?

So far I found just this explanation: http://www.hcs.harvard.edu/~cs134-spring2017/wp-content/uploads/2017/01/section2.pdf. However, that is not clear to me and already the first paragraph is not convincing. I can show her below my concerns about the first paragraph:

I think two notions of neighbors are competing here: neighbors in the graph and neighbors in the ring (they are the same in your first example, but not in the last figure). 'Hops' seem to refer to moves on the ring, not in the graph (again, this makes a difference only on the second example). In addition, when the author mentions "points", I think she refers to elements of $$S$$ only (and then your objection to (d) falls). Does this work any better?