An image post, on the other hand, would typically just contain a single image, with or without a caption/text to go along with it. An audio/video post would be the same but with audio/video added in. Any of these three could use either plugins or standard Embeds to display their content. Titles and authorship might not be displayed for them either, as the content could be self-explanatory.
The quote format is especially well suited to posting a simple quote from a person with no extra information. If you were to put the quote into the post content alone, and put the quoted person’s name into the title of the post, then you could style the post so as to display the_content() by itself but restyled into a blockquote format, and use the_title() to display the quoted person’s name as the byline.
A chat in particular will probably tend towards a monospaced type display, in many cases. With some styling on the .format-chat, you can make it display the content of the post using a monospaced font, perhaps inside a gray background div or similar, thus distinguishing it visually as a chat session.