As mentioned in that post, I'd recommend switching to Joomla's built-in captcha instead of K2's. Unfortunately their captcha is triggered whenever a registration attempt happens, even when it's done using the API and not from a form. We can't 'fix' that issue. You'd have to contact K2 and ask them how to fix the captcha to only perform validation in the spots where they've actually displayed a captcha.
I hope that helps explain,
Alex