We present observations of CaII, ZnII, and CrII absorption lines in 16 DLAs and 6 subDLAs at 0.6 < z < 1.3, obtained for the dual purposes of: (i) clarifying the relationship between DLAs and absorbers selected via strong CaII lines, and (ii) increasing the still limited sample of Zn and Cr abundances in this redshift range. We find only partial overlap between current samples of intermediate-z DLAs (which are drawn from magnitude limited surveys) and strong CaII absorbers: approximately 25% of known DLAs at these redshifts have an associated CaII 3935 line with REW>0.35A, the threshold of the SDSS sample assembled by Wild and her collaborators. The lack of the strongest systems (with REW>0.5A) is consistent with these authors conclusion that such absorbers are often missed in current DLA surveys because they redden/dim the light of the background QSOs. We rule out the suggestion that strong CaII absorption is associated exclusively with the highest-N(HI) DLAs. Furthermore, we find no correlation between the strength of the CaII lines and either the metallicity or depletion, although the strongest CaII absorber in our sample is also the most metal-rich DLA yet discovered, with [Zn/H] ~ solar. We conclude that a complex mix of parameters determine the strengths of the CaII lines, including the density of particles and UV photons in the ISM of the galaxies hosting the DLAs. We find tentative evidence (given the small size of our sample) that strong CaII systems may preferentially sample regions of high gas density, perhaps akin to the DLAs exhibiting molecular hydrogen absorption at redshifts z>2. If this connection is confirmed, strong CaII absorbers would trace possibly metal-rich, H2-bearing columns of cool, dense gas at distances up to tens of kpc from normal galaxies. (abridged)
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