ABRIDGED) We revise the formation of Galactic GCs by adding the detailed chemical composition of their different stellar generations (from 1200 giants in 19 GCs) to their global parameters. We propose to identify as GCs those showing the Na-O anticorrelation, and we classify the GCs according to kinematics and location in the Galaxy in disk/bulge, inner, and outer halo. We find that the LF of GCs is fairly independent of their population, suggesting that it is imprinted by the formation mechanism, and only marginally affected by the ensuing evolution. We show that a large fraction of the primordial population should have been lost by the proto-GCs. The extremely low Al abundances found for the primordial population of massive GCs indicate a very fast enrichment process before the formation of the primordial population. We suggest a scenario for the formation of GCs including at least 3 main phases: i) the formation of a precursor population (likely due to the interaction of cosmological structures similar to those leading to dwarf spheroidals, but residing at smaller Rgc, with the early Galaxy or with other structures), ii) which triggers a large episode of star formation (the primordial population), and iii) the formation of the current GC, mainly within a cooling flow formed by the slow winds of a fraction of the primordial population. The precursor population is very effective in raising the metal content in massive and/or metal poor (mainly halo) clusters, while its role is minor in small and/or metal rich (mainly disk) ones. Finally, we use PCA and multivariate relations to study the phase of metal-enrichment from 1st to 2nd generation. Most of the chemical signatures of GCs may be ascribed to a few parameters, the most important being [Fe/H], mass, and age of the cluster, with the location within the Galaxy also playing some role. ; Comment: 24 pages (+2 pages of bibliography and 5 of Appendix), 19 figures, accepted for publication on Astronomy and Astrophysics