We present new implementations in Method of Moments of two types of second kind integral equations: (i) the recently proposed Electric-Magnetic Field Integral Equation (EMFIE), for perfectly conducting objects, and (ii) the Müller formulation, for homogeneous or piecewise homogeneous dielectric objects. We adopt the Taylor-orthogonal basis functions, a recently presented set of facet-oriented basis functions, which, as we show in this paper, arise from the Taylor's expansion of the current at the centroid of the discretization triangles. We show that the Taylor-orthogonal discretization of the EMFIE mitigates the discrepancy in the computed Radar Cross Section observed in conventional divergence-conforming implementations for moderately small, perfectly conducting, sharp-edged objects. Furthermore, we show that the Taylor-discretization of the Müller-formulation represents a valid option for the analysis of sharp-edged homogenous dielectrics, especially with low dielectric contrasts, when compared with other RWG-discretized implementations for dielectrics. Since the divergence-Taylor Orthogonal basis functions are facet-oriented, they appear better suited than other, edge-oriented, discretization schemes for the analysis of piecewise homogenous objects since they simplify notably the discretization at the junctions arising from the intersection of several dielectric regions.