For Twitter, do people generally use their own name or come up with something catchy if they are an author?
Keep things easy for readers. More important than being clever is that you be found and recognized. Twitter is a discoverability tool in your marketing toolbox, after all. Think of your Twitter handle as an extension of your author brand. With that in mind, go with a close variation of your name so that readers and industry colleagues can find you easily and learn to recognize your name as they scroll their feeds. You can be catchy within that restriction: @ElleDoeWrites, for instance. When folks retweet your catchy tweets, or reply to your great engagements, their followers will easily read your author name in the retweet. The key is to be identifiable as the author you are for the fans you do and will have.