We obtained comprehensive sets of elemental abundances for eleven damped Ly \alpha systems ( DLAs ) at z _ { DLA } = 1.7 - 2.5 .
For nine of them , we accurately constrained their intrinsic abundance patterns accounting for dust depletion and ionization effects .
In Paper I of this series , we showed for three DLA galaxies that we can derive their star formation histories and ages from a detailed comparison of their intrinsic abundance patterns with chemical evolution models .
We determine in this paper the star formation properties of six additional DLA galaxies .
The derived results confirm that no single star formation history explains the diverse sets of abundance patterns in DLAs .
We demonstrate that the various star formation histories reproducing the DLA abundance patterns are typical of local irregular , dwarf starburst and quiescent spiral galaxies .
Independent of the star formation history , the DLAs have a common characteristic of being weak star forming galaxies ; models with high star formation efficiencies are ruled out .
The distribution of the DLA star formation histories shows a trend of finding more galaxies with a star formation history typical of dwarf irregulars with a bursting star formation toward high redshifts , z > 2 .
Only two DLA galaxies ( each at z < 2 ) in our sample of nine objects have a star formation history typical of spiral galaxies .
Since DLAs sample the broad distribution of galaxies at high redshift , this trend indicates that young and less evolved proto-galactic structures with low masses and low star formation rates are more common toward higher redshifts .
This is further supported by the star formation rate and age distributions .
Indeed , all the derived DLA star formation rates per unit area are moderate or low , with values between -3.2 < \log { SFR } < -1.1 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } .
The DLA abundance patterns also require a large spread in ages ranging from 20 Myr up to 3 Gyr .
Enhanced \alpha over iron-peak ratios are associated with young objects having undergone a recent burst of star formation , while solar \alpha over iron-peak ratios are associated with old objects undergoing an inefficient continuous star formation .
The oldest DLA in our sample is observed at z _ { DLA } = 1.864 with an age estimated to more than 3 Gyr ; it nicely indicates that galaxies were already forming at z _ { f } \gtrsim 10 .
But , most of the DLAs show ages much younger than that of the Universe at the epoch of observation .
Young galaxies thus seem to populate the high redshift Universe at z > 2 , suggesting relatively low redshifts of formation ( z \sim 3 ) for most high-redshift galaxies .
The large dispersion in star formation history and age indicates that the DLAs are drawn from a diverse population of galaxies .
The DLA star formation properties are compared with those of other high-redshift galaxies identified in deep imaging surveys with the aim of obtaining a global picture of high-redshift objects .