An author is narrowly defined as the originator of any written work and can thus also be described as a writer.
A writer is a person who uses written words in various styles and techniques to communicate their ideas.
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More broadly defined, an author is "the person who originated or gave existence to anything" and whose authorship determines responsibility for what was created.
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The more specific phrase published author refers to an author whose work has been independently accepted for publication by a reputable publisher, versus a self-publishing author or an unpublished one.
Self-publishing is the publication of any book, album or other media by its author without the involvement of an established publisher.