Alright, people, I had to bump this cause I discovered something. I have no idea why, but as long as I have Proxifier running alone (without PuTTy), I get error 2002.

PS: It happens only when I have Proxifier running alone! If I open PuTTy as well, I can log into FFXIV without problems! If you guys are experiencing error 2002, check if there is any proxy-related program running. In my case, that was the heart of the matter.