We study the problem of nucleation of cracks in gradient damage models used as regularized phase-field model of fracture. We consider specific phase-field models with a linear softening response, with possibly different softening parameters for the spherical and the deviatoric deformation modes. The softening parameters enable the independent adjustment of toughness, stiffness, and peak stress, irrespective of the regularization length-scale. This capability effectively establishes a cohesive-like phase-field model. We exploit a variational formulation to examine the crack nucleation conditions, characterizing stable solutions as directional local minima of the total energy functional. Our analytical results in one dimension reveal intricate scale effects, which emerge from the interaction between the structural scale, the material's cohesive length-scale, and the regularization length. We establish conditions on the material parameters that determine whether the crack nucleation behavior is brittle-like or cohesive-like, offering a clear and classical energetic interpretation for these conditions. These factors influence the conditions for crack nucleation. We establish conditions on the material parameters that determine whether the behavior is brittle-like or cohesive-like, offering a clear and classical energetic interpretation for these conditions. In a two-dimensional setting, we demonstrate that crack nucleation under multi-axial loading presents greater complexity. We show that the volumetric deformation mode can remain stable in the softening regime, regardless of the length-scale effects. Our findings deepen the understanding of the intricate relationship between material softening, the regularization length-scale, the loading mode, and strength. This sheds new light on the numerical results of phase-field models, where solutions with diffused and localized damage can compete.