Fluorescence correlation spectroscopy with total internal reflection excitation (TIR-FCS) is a promising method with emerging biological applications for measuring binding dynamics of fluorescent molecules to a planar substrate as well as diffusion coefficients and concentrations at the interface. Models for correlation functions proposed so far are rather approximate for most conditions, since they neglect lateral diffusion of fluorophores. Here we propose accurate extensions of previously published models for axial correlation functions taking into account lateral diffusion through detection profiles realized in typical experiments. In addition, we consider the effects of surface-generated emission in objective-based TIR-FCS. The expressions for correlation functions presented here will facilitate quantitative and accurate measurements with TIR-FCS.