and optical brightness. Nevertheless, these two populations cover similar ranges of hard X-ray luminosity and absolute K magnitude, thus trace similar levels of gravitational accretion. Consequently, we introduce a new classification based solely on X-ray properties, $HR$ and X-ray luminosity, consistent with the unified AGN model. This X-ray classification uncovers a large fraction of optically obscured, X-ray luminous AGNs missed by the classical optical classification. We find a similar number of X-ray type-1 and type-2 QSOs ($L_{\rm X}$(0.5-10 keV)$>10^{44}$ erg s$^{-1}$) at $z > 2$ (13 sources with unambiguous spectroscopic identification); most X-ray type-1 QSOs are bright, R$ \lesssim 24$, whereas most X-ray type-2 QSOs have R$ \gtrsim 24$ which may explain the difference with the CDFN results as few spectroscopic redshifts were obtained for R$ > 24$ CDFN X-ray counterparts. There are X-ray type-1 QSOs down to $z\sim0.5$, but a strong decrease at $z<2$ in the fraction of luminous X-ray type-2 QSOs may indicate a cosmic evolution of the X-ray luminosity function of the type-2 population. An X-ray spectral analysis is required to confirm this possible evolution. The red colour of most X-ray type-2 AGN could be due to dust associated with the X-ray absorbing material and/or a substantial 