As with "terrestrial", everything on Earth is arguably Terran, i.e. it is just as legitimate to say "of terran origin" as "of terrestrial origin".
Some, such as Hugo De Garis make the distinction that those who want to leave Earth or destroy it may be of terrestrial origin but are not in fact Terran. He defines a "Terran" in political opposition to a "Cosmist" - the latter being those who wish to build artificial intelligence beyond human capabilities and leave Earth.
That distinction seems to be derived from earlier usage of "terran" by many science fiction authors as a synonym for "earthling" or "human" or "terrestrial" - most notably in the context of alien encounters with humans.
Authors seem to most commonly use "terran" or "earthling", sometimes for joke impact, when aliens cannot tell humans from other highly active-and-possibly cognitive terran entities, e.g. dogs, great apes, whales, automobiles. For example, in Star Trek IV - The Voyage Home, an alien ship seeking whales to talk to is causing tremendous harm to Earth, as whales are long extinct and its increases in volume go unheard. Humans aren't recognized as Terrans, obviously, or at least not legitimate ones.
A similar theme was expressed in Visit to a Small Planet, when aliens who visited Earth were convinced it was ruled by cars, with human parasites in them.