TL;DR: According to Book's official ident card, he was simply a retired Alliance Commander.
Officially, the man we know as Derrial Book was not dishonorably discharged - he was quietly "retired" after a massive defeat. Here's what we do know from The Shepherd's Tale:
As commander of the IAV Cortez, Book was responsible for planning & overseeing an operation that was supposed to end The Unification War in a single stroke. While we don't have details, the plan involved launching attacks on multiple targets at once, committing an enormous amount of resources into the attack. Since Book was actually
an operative for the Browncoats, they knew the attack was coming and used it to ambush the Alliance forces.
As a result, countless men & resources were lost, including the total loss of the Alliance Cruiser IAV Alexander and all 4000 people on board. The admiral in charge of the operation considered it the single greatest defeat in the history of The Alliance, and subsequently blamed Book - enough to strip the rank off his uniform and leave him stranded in an escape pod. The entire debacle was an enormous embarassment for The Alliance, so the operation was covered up as much as possible and Book was quietly "retired" from service.
Since everything was "swept under the rug", Book was officially just a retired Alliance commander, and that's what his ident card would identify him as. Only the higher-ups of the Alliance military would know the actual circumstances under which he left the service, but without that information any other Alliance facility or officer would make sure he received the best possible care - as befitting an Alliance war veteran.