In this work, the geometric electromagnetic Particle-in-Cell (PIC) framework, GEMPICX, is extended to solve the quasineutral, fully kinetic Vlasov-Maxwell equations and also the quasineutral, hybrid drift-kinetic Vlasov-Maxwell model, with drift-kinetic electrons and fully kinetic ions. The discretization for both models is performed using structure-preserving finite differences on primal and dual grids. Unknowns are represented as point values, edge integrals, face integrals and volume integrals, on both grids. These mimetic discretizations employ operators that exactly imitate integrated versions of vector calculus identities on discrete spaces. A discrete action principle is derived for both models, taking into account the duality between the grids. The dynamical system for such models does not involve a temporal evolution term for the electric field. It can be obtained implicitly by solving a linear system of equations at each time-step, for both models. This also circumvents the need to obtain electric potentials. For the fully kinetic model, a discretized curl-curl equation is used to implicitly obtain the electric field at every time-step. A Lagrange multiplier is used to maintain the discretized divergence of the current density at machine zero. On the other hand, for the hybrid model, a similarly derived discretized curl-curl equation is used to implicitly obtain the electric field component that is parallel to the background magnetic field. The definition of the current in the drift kinetic model is used to obtain separate equations for the perpendicular component of the electric field. An explicit split time-stepping scheme is used for the fully kinetic model. For the hybrid model, a fully explicit scheme as well as two implicit-explicit (IMEX) schemes are tested. The two models are tested by verifying the various waves obtained from their dispersion relations.