We present spectroscopic observations of the damped Lyα absorber at redshift z = 1.9342 seen in the spectrum of the quasar Q0151+048A. The redshift of the absorber is greater than the redshift of the quasar, so the system resembles the z _abs ≈ z _em damped absorber at z = 2.81 towards the quasar PKS0528-250. We have previously reported the detection of Lyα emission from the latter absorber, one of only two damped absorbers for which Lyα emission has unambiguously been detected. The resemblance between the PKS0528-250 and Q0151+048A systems is made closer by the detection of a weak emission feature in the trough of the Q0151+048A absorber. This leads us to consider whether these z _abs ≈ z _em DLA absorbers are different objects to the intervening DLA absorbers. Two possibilities are examined and rejected. Firstly the Q0151+048A and PKS0528-250 z _abs ≈ z _em absorbers appear to be unrelated to the intrinsic absorbers (i.e. gas close to the quasar nucleus, ejected by the quasar), as intrinsic absorbers are of higher metallicity, have higher ionisation parameter, and show complex absorption profiles. Secondly these two DLA absorbers cannot be equated with the gaseous disks of the quasar host galaxies, as the absorber redshifts differ significantly from the quasar systemic redshifts. It is likely, then, that intrinsically the z _abs ≈ z _em DLA absorbers are the same as the intervening DLA absorbers, so that peculiarities in some of the z _abs ≈ z _em absorbers can be ascribed to their different environment i.e. proximity to the quasar, or membership of the same cluster as the quasar. We point out that the proximity effect may play some rôle, by reducing the Lyα-forest line blanketing of any Lyα emission line from z _abs ≈ z _em absorbers.