LTI system and initial conditions
The system is described by a linear difference equation with constant coefficients and as such, it is described in the same way as a linear time-invariant system. It is just the non-zero initial condition that makes the system non-linear, at least according to the common definition of linearity in system theory (homogeneity and additivity).

Such a system with non-zero initial conditions is also referred to as incrementally linear. An incrementally linear system responds linearly to changes in the input. For this reason it is very common, and it also makes a lot of sense, to discuss it in the same chapter as general discrete-time LTI systems.

As a final note, in practice we're mainly interested in stable systems. For such systems, the influence of non-zero initial conditions becomes negligible after some time, and, consequently, the system will practically behave as an LTI system after the transient caused by the non-zero initial conditions has died out.