Earlier quoted context omitted.
I don't think JPEG wants to produce an image indistinguishable from the original. It wants to reduce space usage without distorting "too" much. Failing to reduce space usage would be considered a "failure" of JPEG, just as much as distorting too much.
JPEG relies of the limitations of human vision to make an image largely indistinguishable from the original. It specifically throws information away that we are less unlikely to notice. So yes, a good JPEG should be indistinguishable (to humans) from the original. Obviously the more you turn up the compression the harder that is.
It's not quite that straight-forward, though, in that there are two competing goals: Small size and looking as similar as possible to the original. We're explicitly willing trade accuracy for size. How much depends on the use, but sometimes we're willing to trade away so much quality that the artefacts are plainly visible. And we're willing to trade more accuracy for size when the artefacts doesn't distract. For some uses compression artefacts are better than misleading about the original, but for other uses, misleading changes would be preferable as long as they don't give fewer noticeable artefacts for a given size.