So, I think I understand it a bit better now. The EN difference should only be used to discuss the extent to which the electrons are "shared" among the two bonded atoms. With larger EN differences, the bond displays more "ionic" character, meaning that there is a stronger polar bond. With smaller EN differences, the atoms (whether an ionic, e.g. CuCl2, or covalent compound, e.g. CCl4) display more "sharing" and a less polar bond.
So, CuCl2 (as initially expected) is an ionic compound, though the bonds are not as polar as something like LiF. Am I correct?