Private vs public vs formal vs material heretics
From all I've read, it seems that the following would be correct definitions of all of the above:
Private: Internal, occult, kept hidden in the heart. No one (except, perhaps, his confessor) knows he doubts or rejects any Catholic doctrine. He keeps that doubt hidden from all. He remains a member of the Church.
Public: External, proclaimed. Others know he doubts or rejects a Catholic doctrine. He does not keep his heretical beliefs secret. It doesn't matter how many people he tells, the fact is that he has made it know to others that he disbelieves a Catholic doctrine. He removes himself from the visible unity of the Church and is no longer a member of the Church.
Formal: Must be public. Even though he knows what the doctrine of the Church is, he rejects that doctrine. He is not simply in error of fact, he knows what the facts are. He removes himself from the visible unity of the Church and is no longer a member of the Church.
Material: Could be public or private. He believes something that is contrary to Catholic doctrine but that belief is either unintentional because he has misunderstood or was taught incorrectly. However, he is docile to the magisterium and will correct his beliefs upon learning the truth. He may simply be in error of fact. He remains a member of the Church but, should he learn the truth and obstinately reject it, his heresy becomes formal and he removes himself from the visible unity of the Church and is no longer a member of the Church.
Note: Just as Christ noted that if they are blind they have no guilt but because they say they see their guilt remains, one cannot reasonably consider a cleric to be ignorant of the faith.