We build a sample of O VI absorption systems in the redshift range 2.0 $\lesssim z \lesssim$ 2.6 using high spectral resolution data of ten quasars from the VLT-UVES Large Programme. We investigate the existence of a metal-rich O VI population and define observational criteria for this class of absorbers under the assumption of photoionisation. The low temperatures of nearly half of all O VI absorbers, implied by their line widths, are too low for collisional ionisation to be a dominant process. We estimate the oxygen abundance under the assumption of photoionisation; a striking result is the bimodal distribution of [O/H] with median values close to 0.01 and 0.5 Solar for the metal-poor and metal-rich populations, respectively. Using the line widths to fix the temperature or assuming a constant, low gas density does not drastically change the metallicities of the metal-rich population. We present the first estimate of the O VI column density distribution. Assuming a single power-law distribution, $f$(N) $\propto$ N$^{-\alpha}$, yields $\alpha \sim 1.7$ and a normalisation of $f$(N) $=2.3\times 10^{-13}$ at log N(O VI) $\sim$ 13.5, both with a $\sim$30% uncertainty. The value of $\alpha$ is similar to that found for C IV surveys, whereas the normalisation factor is about ten times higher. We use $f$(N) to derive the number density per unit $z$ and cosmic density $\Omega_{\rm b}$(O VI), selecting a limited column density range not strongly affected by incompleteness or sample variance. Comparing our results with those obtained at $z\sim0.1$ for a similar range of column densities implies some decline of $dn/dz$ with $z$. The cosmic O VI density derived from $f$(N), $\Omega_{\rm b}$(O VI)$\approx (3.5\pm ^{3.2}_{0.9}) \times 10^{-7}$, is 2.3 times higher than the value estimated using the observed O VI sample (of which the metal-rich population contributes $\sim$35%), easing the problem of missing metals at high $z$ ($\sim$ 1/4 of the produced metals) but not solving it. We find that the majority of the metal-rich absorbers are located within $\sim$ 450 km s$^{-1}$ of strong Ly-$\alpha$ lines and show that, contrary to the metal-poor absorbers, this population cannot be in hydrostatic equilibrium. All of the O VI absorber properties imply that there are two distinct populations: metal-poor absorbers tracing the intergalactic medium and metal-rich absorbers associated with active sites of star formation and most probably linked to galactic winds.