I agree that simple examples help build intuition. That’s why I would do it first using the bivariate normal copula likelihoods and check that the vine version produces the same results. Then you can write a non-normal bivariate copula likelihood and see what happens when you change one of them.
Check out pages 9 and 10 of this
I think that makes it pretty clear on how the decomposition is happening. Page 9 would require you to condition variables 1 and 3 on variable 2, choose a density for that relationship, and then fit a copula. C/R/D vine is just about how to choose the decomposition.