A coherent topos is a sheaf topos with a site of definition that is a coherent category with coherent coverage.
Notice that the terminology is slightly confusing: every topos is a coherent category.
finitely complete category, cartesian functor, cartesian logic, cartesian theory
regular category, regular functor, regular logic, regular theory, regular coverage, regular topos
coherent category, coherent functor, coherent logic, coherent theory, coherent coverage, coherent topos
geometric category, geometric functor, geometric logic, geometric theory
Section D3.3 of