Doujin/doujinshi is a tough thing to translate, since it can actually be a large variety of things. Though manga are what people are most familiar with (doujinshi), there's also doujin games/software, doujin music, doujin novels, and more.
While I can understand the 'fanzine' translation, my feeling of a 'zine is that it's like a magazine, which simply includes info maybe some pictures, articles, etc. about whatever group or thing you're passionate about. Look up old Star Trek, band, and toy fanzines from the 90s and earlier to see what I'm talking about. I don't think it catches the meaning of what's being described.
At the end of the day, a 'doujinshi' is a fan-made, unlicensed comic based off of a series. Unlicensed comic sounds lame, but I think it's closer to the meaning.