A telefax is a document that transmits between two fax (telefax) machines.
It can also be a verb meaning the action of transmitting a document between two fax machines.
According to the Merriam Webster dictionary, “telefax” is the same as the second definition of “facsimile”:
: a system of transmitting and reproducing graphic matter (such as printing or still pictures) by means of signals sent over telephone lines
Top synonyms of “fax” according to Thesaurus.com are “facsimile,” “facsimile machine,” “telefacsimile,” and “telephotograph.”
Merriam-Webster lists the origin of the word “telefax” having a first known use in 1943.
A search of Google books shows the use of the term in a 1944’s Volume 51 of the Public Utilities Reports by Lawyers Cooperative Publishing Company.
Western Union’s American Telegraphy after 100 years (published in 1944 as well) includes the line “… although the extension of the telefax program had to interrupted on account of the war…” This line indicates that the term was in use before 1942.
A facsimile can mean the same as the other terms, and it can also mean a copy or exact reproduction in the sense of a manuscript or piece of artwork.