The morphologies of massive galaxies from z~3 - Witnessing the 2 channels of bulge growth

[abridged] We quantify the morphological evolution of z~0 massive galaxies ($M*/M_\odot\sim10^{11}$) from z~3 in the 5 CANDELS fields. The progenitors are selected using abundance matching techniques to account for the mass growth. The morphologies strongly evolve from z~3. At z<1, the population matches the massive end of the Hubble sequence, with 30% of spheroids, 50% of galaxies with equally dominant disk and bulge components and 20% of disks. At z~2-3 there is a majority of irregular systems (~60-70%) with still 30% of spheroids. We then analyze the SFRs, gas fractions and structural properties for the different morphologies independently. Our results suggest two distinct channels for the growth of bulges in massive galaxies. Around 30-40% were already bulges at z~2.5, with low average SFRs and gas-fractions (10-15%), high Sersic indices (n>3-4) and small effective radii ($R_e$~1 kpc) pointing towards an early formation through gas-rich mergers or VDI. Between z~ 2.5 and z~0, they rapidly increase their size by a factor of ~4-5, become all passive but their global morphology remains unaltered. The structural evolution is independent of the gas fractions, suggesting that it is driven by ex-situ events. The remaining 60% experience a gradual morphological transformation, from clumpy disks to more regular bulge+disks systems, essentially happening at z>1. It results in the growth of a significant bulge component (n~3) for 2/3 of the systems possibly through the migration of clumps while the remaining 1/3 keeps a rather small bulge (n~1.5-2). The transition phase between disturbed and relaxed systems and the emergence of the bulge is correlated with a decrease of the star formation activity and the gas fractions. The growth of the effective radii scales roughly with $H(z)^{-1}$ and it is therefore consistent with the expected growth of disks in galaxy haloes.