Numerical modeling of structures made of softening materials remains a complex challenge due to the loss of well-posedness and objectivity inherent to continuum local damage formulations. To remedy this limitation, numerous non-local regularization techniques introducing an internal length scale have been proposed. Among these, the variational Phase-Field approach to fracture has emerged as a robust framework. Damage evolution is formulated as the minimization of a total energy functional composed of elastic and dissipation energies, with non-locality introduced through a gradient-based regularization term. More recently, an alternative strategy known as the Lip-Field formulation has been proposed. This approach preserves a fully local energy functional and enforces regularization by constraining the damage field to satisfy a Lipschitz condition. The present study provides a comparative assessment of Phase-Field and Lip-Field formulations with respect to their regularization properties and computational efficiency. To ensure meaningful comparison, cohesive-like damage models are employed, allowing both approaches to reproduce equivalent fracture responses at the structural level. Several finite element solution strategies are investigated, including standard and alternative solvers based on either one- or two-mesh discretizations within an alternating minimization framework. The solvers are evaluated by analyzing the convergence of both the staggered solution procedure and the minimization algorithm used to compute the regularized damage field. Finally, quantitative error measures are introduced, and their behavior under mesh refinement is examined to assess the accuracy and robustness of the Phase-Field and Lip-Field approaches.