All international carriers that I have flown on, prohibit smoking on board flights and all have added specifically e-cigarettes to the announcement/safety video to explicitly state that these are also forbidden.
The US Department of Transportation prohibits the use of e-cigarettes and in addition, prohibits the charging of these devices on-board - which is also recommended by ICAO:
U.S. Department of Transportation Pipeline and Hazardous Materials
Safety Administration today issued an interim final rule (IFR) to
prohibit passengers and crewmembers from carrying battery-powered
portable electronic smoking devices (e.g. e-cigarettes, e-cigs,
e-cigars, e-pipes, personal vaporizers, electronic nicotine delivery
systems) in checked baggage and prohibit passengers and crewmembers
from charging the devices and/or batteries on board the aircraft.
“We know from recent incidents that e-cigarettes in checked bags can
catch fire during transport,” said Transportation Secretary Anthony
Foxx. “Fire hazards in flight are particularly dangerous. Banning
e-cigarettes from checked bags is a prudent safety measure.”
On January 22, 2015, the Federal Aviation Administration issued a
Safety Alert for Operators recommending that air carriers require
their passengers to carry e-cigarettes and related devices exclusively
in the cabin of the aircraft. Also, on June 9, 2015, the
International Civil Aviation Organization (ICAO) published an addendum
to the 2015-2016 ICAO Technical Instructions for the Safe Transport of
Dangerous Goods by Air prohibiting the carriage of e-cigarettes in
checked baggage and restricting the charging of these devices while on
board the aircraft.
Passengers may continue to carry e-cigarettes for personal use in
carry-on baggage or on their person but may not use them on flights.
The Department’s current regulatory ban on smoking of tobacco products
on passenger flights includes the use of electronic cigarettes.
Nevertheless, to prevent passenger or crewmember confusion, the
Department has proposed to amend its existing airline smoking rule to
explicitly ban use of electronic cigarettes aboard aircraft.
In effect this means that all US-flagged carriers prohibit smoking of e-cigarettes on board; all carriers prohibit charging of these devices on board; and could also possibly mean that any carrier flying into the US should also abide by these rules.
For Europe, EASA also has a similar safety information bulletin based on the ICAO bulletin which also has the same restrictions.
I should note that these restrictions go towards charging of the device as it has a heated element which is a fire hazard. As far as I understand charging of the device is an integral part of the entire smoking experience - so it would make sense that smoking them would also be forbidden.
Finally, I don't know anyone who would assume vapestick (a manufacturer of e-cigarettes) is the authority on where they should be used. Even if you ignore the bias, it is a very, very, very bad idea to smoke in the toilet of an aircraft. It is the fastest way to get a fine, be banned from the airline, and possibly put in jail.