Definitions of "fake" and "forgery"
A faked document is an original document that has been altered to trick people into believing it is rare and valuable. For example, the creator falsely states the age or origin of the document to deceive people into believing it is historically significant and therefore of considerable value. A recently created will or diary attributed to a person who never created such a document would be a fake.
A forgery is an imitation or a copy that is represented as the original. It is created to trick people into believing it is the real thing. An exact copy of a painting by Tom Thomson, falsely said to have been painted by Thomson, would be a forgery.