How can I distinguish whether I have dark under eye circles or hyperpigmented skin under my eyes?  

Watch for changes. "dark circles under eyes" just means the coloring of the lower eyelids and skin below the lower lids is darker than other parts of the face. Dark circles can be from increased pigment in the skin, more blood in the skin, thinner skin so the blood in the skin shows up more easily, and any combination of the three. The coloring that changes from hour to hour, or day to day, is probably not pigment.