Upvote:2
Did Adam Weishaupt's Bavarian Illuminati know of the Spanish group? Is there anything in the writings of the Bavarians which resembles the ideas of the Spanish? If not, then this seems to be simply a coincidence of names.
There's also the question of which Illuminati mythos you're asking about. There are at least two:
One is the line through Freemasonry and related groups, such as the Ordo Templi Orientis. The important factor for distinguishing this "authentic" mythos is that it has demonstrable sources before 1975.
The other line started in 1975 with The Illuminatus! Trilogy, by Robert Shea and Robert Anton Wilson. They started with some basic facts about the Bavarians and made up a lot of plausible-seeming fiction on top of it. They were thoroughly familiar with Discordianism, which is a postmodern joke disguised as a religion (or a religion disguised as a postmodern joke - it's hard to tell) and applied its principles thoroughly in their writing.
In particular, Operation Mindf*ck is an important part of Discordianism. That requires the attribution of "all national calamities, assassinations, or conspiracies" to the Bavarian Illuminati, in a deliberate attempt to spread paranoia. This was extremely successful, and all modern Illuminati conspiracy theorists are their victims.