IANAL, etc., but I believe there's lot's of good legal precedent around the legitimacy of FAX, and a lot of doubt and uncertainty about other electronic transfers of documents. So FAX is the safe choice.
In the heyday of modems perhaps 20 years ago, this was "normal" if you already had a scanner, you could send faxes using most higher end modems, like 14.4 and faster. So this has been technologically obsolete as an authentication system for a couple decades.
I believe the belief comes from the bad old days of itemized long distance billing. AT&T will send a bill to me proving I sent "something" to them so if they claim I didn't, then produce what they claim I sent. Sorta registered mail service, kind of. But itemized long distance billing is dead.
For HN readers too young to know what itemized long distance billing is, before the modern era of "all minutes are free" you'd seriously, no kidding, get a paper bill listing all your interLATA phone calls and what they cost.
Wash DC 202 123-4567 5 minutes daytime rate subtotal $10
The kind of thing seen above.