The intermittent damage evolution preceding the failure of heterogeneous solids is well-described by robust scaling relations. Origin of such statistics is often attributed to the strength of material disorder. Yet, the acoustic emissions during failure of weakly disordered solids also exhibit robust scale-free statistics. Here, we numerically examine the precursory damage evolution considering an additional parameter, the damage hardening of the mesoscopic elements to interpret the microfracturing processes at lower length-scales. Interestingly, it allows a critical interpretation of precursory damage activity even for a case of weak disorder. We then derive a quasi-brittleness phase diagram showing that brittle and ductile-like failure of heterogeneous solids belong to different universality classes, reminiscent of the percolation and the depinning transitions. Our findings shed light on the range of Omori law exponents reported in literature as corresponding to the brittleness of the damaging solid.