I have new evidence on the Disciple hat. I conducted a scientific test on 6 different sites. Site A = followed 5 posts, Site B = followed 6 posts, Site C = followed 7 posts, Site D = followed 8 posts, Site E = followed 9 posts, Site F = followed 10 posts. No other actions were done on those sites.
After several minutes, I received the Disciple hat on site D, E, and F. Those were the sites that I had followed 8, 9, and 10 posts.
This means that the trigger for the Disciple hat could either be "follow 8 posts" or something else that involves votes, like "follow 6 posts with a score of 1 or more"