The official name of the competition is SUPER LEAGUE.
As Flamin' says above the company name is Super League (Europe) Ltd., it was registered at the same time time as the Aussie Super League, hence the brackets.
Engage are the sponsor and thus the name Engage Super League is regularly used, similar to the Telstra Premiership. However this is not a lifetime thing, sponsors change and the competition used to be called Tetley Super League.
With the company name being having "Europe" in the title but in brackets, as well as there being Paris, Catalans and now Crusaders in the competition some fans have called it Super League Europe - probably to make the competition sound bigger than it is, to make it into the supra-national top-fight competition for the whole continent, or in some cases to differentiate between the 1997 Aussie SL. However it is never officially been called Super League Europe or SLE.
With the Engage name (Engage Mutual are an insurance firm btw) it was regularly shortened to ESL which meant that it is sometimes called, again mistakenly, European Super League by fans.
The competition is not called the English Super League, it never has been and it never will be. It is the name still used by stupid moronic Australian journalists. It was first coined during the Super League war in Australia, even though there was a Paris team I think the name was first used before teams were decided, so journalists presumed it would just be an English competition.
It astounds me though that a rugby league forum, with actual league fans, has called this forum the English Super League.