The best way can simplify this for you is to tell you to use an article everytime, however in the first sentence the article would be the. They may request your name as the writer. - The listener was the writer.
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Thanks for the reply. 'The' made it clearer and more natural, but I actually came across the first sentence in a book (more formal writing, in business proposals or publishing materials). I am seeing these patterns a lot, so there must be a good explanation why we use it this way.
Good day!