The reasons you mention are important and many people have a blend of them as their reason, which is probably a good thing. Writers have a something inside them that motivates them to do it. Perhaps no one else ever sees their work, perhaps it always stays in their head, but they write.
I love that writing allows me to express myself precisely if I work at the process. When I fail to do so and make mistakes or do not clearly write what I mean I frustrate myself. Using the frustration to renew commitment to working toward preciseness diminishes that feeling.
Words are fun stuff when used creatively and the enjoyment derived from using them that way is worth the work to me. Even if the topic is serious working creatively with the words used to express myself is very satisfying.