Is the statement that it is obligatory to khallil (run one's fingers through) both thin and thick hair during ghusl (ritual bath) correct? And how is water to be delivered to the surface of the skin?

The statement attributed to the book, claiming that washing the hair without letting water reach the scalp is sufficient for ghusl (definition), is incorrect. Indeed, the jurists have agreed upon the obligation of letting water reach the skin during ghusl, whether the hair is thin or thick. This is achieved by running water through the hair until it is presumed that the water has reached the roots of the scalp.