I've argued in the past that self-published authors mistakenly call themselves "independent" or "indie" authors, misusing the term traditionally used by independent presses. Indie presses are not self-publishing businesses; they are tradional publishers. Indie presses, also called small presses, use the same submission process used by Big Six publishers, but without the financial resources of the bigger publishers, they are limited in the number of books they can produce and the advances they can pay their authors. However, their limits (e.g., nonprofit status through association with a university) also seem to free them to take chances on books that might not be as commercially successful as those championed by larger publishers.
That said, I'm rethinking my argument based on the article "What Is An Indie Author?" by Orna Ross of the Alliance of Independent Authors. Based on Ross's careful discussion of the term "indie author," I'm thinking that the word is flexible enough to accommodate both authors published by indie presses and self-published indie authors. I do, however, think it is important to keep that distinction alive, because I think it would be a shame if people confused the contributions of indie publishers with those of self-published authors. Based on what I've seen, the majority of self-published work still does not meet the standards of work published traditionally, including work published by indie presses.
To date, I have seen no substitute in self-publishing for the processes a book goes through at a traditional publisher. That's because traditional publishers large and small invest substantially in the books they produce. They have a system in place for editing, design, and production, and these processes are carried out by professionals paid professional wages. These elements make the difference in quality between traditionally published books and self-published books. Until I can no longer tell which books (print or electronic) have been published traditionally and which have been self-published, I'll continue to push the distinction between indie publishers and indie authors.