We present new general relativistic simulations of stellar core collapse to a proto-neutron star, using various microphysical equations of state as well as a description of deleptonization. We show that for a wide variety of rotation rates and profiles the gravitational wave burst signals from the core bounce are of a generic type, known as Type I in the literature. We identify and individually quantify the micro- and macrophysical mechanisms leading to this result. Such a generic type of signal templates will likely facilitate a more efficient search in current and future gravitational wave detectors of both interferometric and resonant type.