How do I get Captcha to reappear in my registration form if an error was detected after submission?
It's in module settings. Under Persistence , you can set CAPTCHA to appear:


always,
once per form,
once per session.


It used to default to always, but as far as I remember there was an issue to change default. Probably now it defaults to once per form or once per session. It should be enough to manually change it to always.