Independent comics, of which self-published comics are a subset, can be as individual and idiosyncratic as independent books that avoid the major publishing house, independent magazines that carry no advertising, and independent newspapers that provide a contrary approach to stories.
You can guess that Dog Boy on Cat Head Comics is an independent publication. But it need not necessarily be those sheets that veer from the subject matter of the majors, that earn the right to be called independent. Not all independent comics are alternative comics or underground comics but they are both classed as independent by their very nature.
A creator or small creative team with an idea for a matricide take it to a mainstream company. It doesn't fit with their focus at the time, so the next step is contacting an independent concern to see if they're interested. They might be, that's just the thing - there's no requirement that the treatment is ironic or irreverent, satiric or sadistic. It could be written up as a straight 'killing mummy' story or series. As long as there are no shareholders or multimedia parent companies involved, the imprint it comes out on would be classed as independent.