In the literature, sometimes one reads that decoherence is due to the coupling of the system to the external environment, and sometimes one reads that it is due to coarse graining over the microscopic degrees of freedom. Are these two different cases of decoherence, or is one more fundamental than the other?
The more conventional way is to describe decoherence as being due to the "coupling to the environmental degrees of freedom" that are traced over. However, the "environmental degrees of freedom" may also include geometrically internal degrees of freedom of a physical system such as a cat – unmeasurably complicated correlations in the properties of the individual atoms.
Because the degrees of freedom you may track – e.g. whether or not a cat is alive – become entangled with many degrees of freedom you can't track – e.g. a jungle of phonons propagating through the cat – you may derive that the density matrix for the latter becomes quickly diagonalized. In this picture, the phonons' degrees of freedom would be considered "environment" by those who say that decoherence is due to the coupling to the environment. The other group wouldn't call it the environment. Instead, it would refer to coarse-graining in which all microstates of a cat which are alive, regardless of the state of the phonons etc., are clumped together. One would derive the appearance of the non-pure density matrix via another approximation but the qualitative outcome would be the same: decoherence.