We have performed a new abundance analysis of Carina red giant (RG) stars from spectroscopic data collected with UVES (high spectral resolution) and FLAMES/GIRAFFE (high and medium resolution) at ESO/VLT. The former sample includes 44 RGs, while the latter consists of 65 (high-resolution) and ~800 (medium-resolution) RGs, covering a significant fraction of the galaxy's RG branch, and red clump stars. To improve the abundance analysis at the faint magnitude limit, the FLAMES/GIRAFFE data were divided into ten surface gravity and effective temperature bins. The spectra of the stars belonging to the same gravity and temperature bin were stacked. This approach allowed us to increase the signal-to-noise ratio in the faint magnitude limit (V≥ 20.5 mag) by at least a factor of five. We took advantage of the new photometry index cU,B,I introduced recently as an age and probably a metallicity indicator to split stars along the red giant branch. These two stellar populations display distinct [Fe/H] and [Mg/H] distributions: their mean iron abundances are -2.15 ± 0.06 dex (σ = 0.28), and -1.75 ± 0.03 dex (σ = 0.21), respectively. The two iron distributions differ at the 75% level. This supports preliminary results. Moreover, we found that the old and intermediate-age stellar populations have mean [Mg/H] abundances of -1.91 ± 0.05 dex (σ = 0.22) and -1.35 ± 0.03 dex (σ = 0.22); these differ at the 83% level. Carina's α-element abundances agree, within 1σ, with similar abundances for field halo stars and for cluster (Galactic and Magellanic) stars. The same outcome applies to nearby dwarf spheroidals and ultra-faint dwarf galaxies in the iron range covered by Carina stars. Finally, we found evidence of a clear correlation between Na and O abundances, thus suggesting that Carina's chemical enrichment history is quite different from that in the globular clusters. Based on spectra retrieved from the ESO/ST-ECF Science Archive Facility and collected either with UVES at ESO/VLT (065.N-0378(A), 066.B-0320(A), P.I.: E. Tolstoy) or with FLAMES/GIRAFFE-UVES at ESO/VLT (074.B-0415(A), 076.B-0146(A), P.I.: E. Tolstoy; 171.B-0520(A)(B)(C), 180.B-0806(B), P.I.: G. Gilmore).Full Tables 2-6, and 8 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/580/A18