I believe this article might help explain the differences Difference Between Author and Writer | Difference Between | Author vs Writer

Authors are always writers; not all writers are authors.

Many people believe that an "author" is a writer of books where a writer is someone who writes for other publications, such as magazines or newspapers.

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You're a writer because you write. You're an author of something that people actually read. That's my definition.

You can be a writer on a TV show, for example, but you can't call yourself an author. You may have authored a book, but you don't have to have "written" it yourself. Does it make sense?