It gets in the way. I'd like to have it disappear when I start typing, then reappear if I move it. I've heard of
unclutter but it sounds like it hides the cursor after it's been "idle" for a while, which isn't what I want--I only want to hide when I'm typing. How could I do this?
You can use xbanish, it does the job. Install it and run it simply like that: