Do people really still have parts of a nictitating membrane?

No. Fully developed nictitating membranes are found in birds, reptiles, amphibians, fish, and mammals, but are rare in primates. In humans, the plica semilunaris (also known as the semilunar fold) and its associated muscles are thought to be probably homologous to the nictitating membranes seen in some other mammals and other vertebrates. In most primate species, a plica semilunaris is present.