Fax is short for “facsimile” when used in the context of electronically transmitting a document between fax machines.
The official definition of “facsimile” is a copy or reproduction of an item (typically a work of art or a document).
A reproduced work of art is called a facsimile, as is a reproduced map. Neither of these are called faxes.
A document electronically transmitted between fax machines is called a fax. It can also be called a facsimile.
Sources and more information:
Wikipedia – Fax
Adams on Contract Drafting – A Gripping Issue of Notice-Provision Technology: “Telecopier,” “Facsimile,” or “Fax”?