Cracking in concrete typically starts in the immediate vicinity around the aggregates, i.e., in the region of the so-called interfacial transition zones (ITZs), but the process is still not fully understood. Notably, crushing of concrete in compression results in fragments with interesting aggregate surface textures. Part of the aggregate surfaces is cleanly separated from the ITZ, while another part of the aggregate surfaces remains covered with a thin layer of cement paste. This suggests two different types of failure: ITZ-aggregate separation and ITZ failure; which we here study based on the continuum micromechanics approach of the companion paper (part I). It provides access to both traction vectors acting on aggregate surfaces and three-dimensional stress states within representative ITZ volumes for loading states below the elastic limit of concrete. When inserting these microtractions and microstresses into Rankine-type strength criteria for the aggregate-ITZ interface and for the ITZ, respectively, the micromechanics model allows for upscaling this microscopic failure behavior to concrete-level criteria for crack onset. Comparing the latter to corresponding experimental results, reveals that under tension-dominated loading both ITZ failure and ITZ-aggregate separation appear to be realistic, while under compression-dominated loading ITZ failure appears as the more likely mechanism. Also, comparing model and experiments shows that the ITZ-aggregate separation strength amounts to at least half of the internal ITZ cohesion strength, but may be much larger than the latter.