Between fax, facsimile, telefacsimile, and telecopier, your best bet is fax.
“Telecopier” is a brand name, so it’s best not to use it to refer to all makes of machine.
And fax is a better option than facsimile and telefacsimile, with respect to both the machine and that which it transmits. Garner’s Modern American Usage says that fax “is now all but universal, in the face of which facsimile transmission is an instant archaism – and a trifle pompous at that.” It suggests that fax “is now perfectly appropriate even in formal contexts.”
Don’t write fax in all capitals – it’s not an acronym.
The fax machine is heading toward obsolescence, but you can expect that contracts will continue to refer to faxing for a good long while. After all, Telex appears to be all but dead as a means of communication, but plenty of contracts still refer to giving notices by Telex.