I would like to know what's the difference between cis-regulating elements and DNase I–hypersensitive sites, in order to produce a meaningful segregation of non coding elements affecting gene expression.
Cis-regulatory elements are simply DNA regions upstream or downstream of a gene that can affect its expression (basically they have to be in the same chromosome).
DNAse-I hypersensitive sites (DHS) are regions of chromatin that get digested during the DNAse treatment because they are exposed i.e. not protected by a protein (complex). The protein complex can either be a nucleosome or a transcription factor. It should be noted that a DNA region can be DNAse insensitive also because of its conformation.
Usually the cis-regulatory elements exert their effects by serving as a binding site for a transcription factors (or chromatin remodellers). If the proteins bind at these sites then these regions would be protected from DNAse digestion thereby making them not-DHS. However, it is known that DNA elements can act in trans as well (see papers on 3C- chromosome conformation capture, and its variants). Moreover, whether a site is protected from DNAse or not does not tell anything about its function.