There is a separate Klingon word Human which refers to a human.
Theoretically, I suppose, a human could have grown up on a Lunar base or on Mars; he'd still be a Human but I'm not sure whether he would still be a tera'ngan.
That said, we usually accept "human" as a translation of tera'ngan, since the distinction is a little philosophical, so if it's not accepted here, it probably should, for consistency if nothing else.
Could a Terran also be a non-human who grew up on Earth? I'd imagine so, but unsure if this has been addressed in canon.
In every context the word tera'ngan has been used, it's been synonymous with "human." Humans will walk into Klingon shops and the Klingon calls him a tera'ngan. I would assume that in general circumstances Klingons don't care about the technical difference and call all humans Terrans. And when the sniveling human says tera'ngan jIHbe'; not tera' vISuchpu' "I'm not a Terran; I've never visited Earth," the nearest Klingon shoots him for being an irritant.
Naturally if you specifically talking about the homes of beings on different planets the listening Klingon would care about getting the terms right. In such a circumstance a tera'ngan is different than an 'orghenya' Dabbogh Human but might include a tera' Dabbogh vulqangan. (Notice there's no way to distinguish between an inhabitant of Vulcan and a member of the Vulcan species. Likewise with verengan, romuluSngan, and qarDaSngan.)