I had thought this definition was clear and correct, until I asked about the third nullification of Islam: in failing to declare a disbeliever a kafir (i.e., takfir), one has disbelieved. Uma's comment on this question indicates I have a misunderstanding.
Obviously, it's undesirable for unqualified Muslims to go around accusing each other of disbelief, and Sahih Muslim 60 a (among other ahadith, and possibly even Qur'an 4:94) strongly discourages that. III-AK-III's answer to the above question says that only by "a knowledgeable body of a certain level of scholars (mujtahid levels)". This suggests takfir is something like a fatwa.
Consequently, it seems there's more to takfir than simply declaring someone a disbeliever. And hence my question:
Question: What exactly is takfir?
Perhaps there's a distinction between how it should be done, and how it's often done.