A whole series of complementary studies have been performed on the same single nanowire containing a quantum dot: cathodoluminescence spectroscopy and imaging, micro-photoluminescence spectroscopy under magnetic field and as a function of temperature, and energy-dispersive x-ray spectrometry and imaging. The ZnTe nanowire was deposited on a Si3N4 membrane with Ti/Al patterns. The complete set of data shows that the CdTe quantum dot features the heavy-hole state as a ground state, although the compressive mismatch strain promotes a light-hole ground state as soon as the aspect ratio is larger than unity (elongated dot). A numerical calculation of the whole structure shows that the transition from the heavy-hole to the light-hole configuration is pushed toward values of the aspect ratio much larger than unity by the presence of a (Zn, Mg)Te shell, and that the effect is further enhanced by a small valence band offset between the semiconductors in the dot and around it.