There are different 'types' of stunting. for instance, take a goldfish and put it in a 55 Gallon, underfeed it. It will not achieve what is considered proper growth, and is therefore stunted. It will appear emaciated, be vulnerable to disease and be unhealthy. If this is not corrected in time, and occurs from a young age, the fish although fed well for the rest of its life will like always carry the physical consequences, like a malnourished human child. Its growth rate throughout its mass may not be even, giving it the appearance of lumpy deformity.
Take a fish which is kept in a cramped tank leaving it no room to maneuvre normally; this can result in skeletal deformities, which will inhibit growth, but not prevent it, and impact on internal organs. this is probably closest to the classic perception of what stunting is and while it is not necessarily that the fish stops growing its skeleton while continuing to develop its internal organs, it is somewhat close, in that the deformed skeleton arising form cramped conditions impacts negatively on development/performance of internal organs, often swim bladder.
take fish which are kept overstocked and underfiltered. the release of growth inhibitor hormones by dominant fish will inhibit growth, and again lead to stunting of physical size, but as long as sufficient food is eaten, there being no skeletal abnormality, the fish may not appear deformed except in size. the underfiltering will of course impact on health in other ways.
your classic Oscar in a 20G for life kept by someone who doesn't know better is probably going to be overfed, so big bulgy belly, underfiltered and insufficient water changes, so disease problems from this, and if achieving any real size may encounter the skeletal deformities...hence the 'stunted' fish.
There is a real difficulty in isolating the cause of the illness and deformities in a stunted fish because there are a) different causes of stunting and b) stunting will only over occur to a fish which is being raised in a poor environment and this will in and of itself cause problems.
The 'Organs keep growing when stunted' hypothesis is not in my opinion 'mythbusted' but is an over simplified way of describing symptoms/deformities often present in 'stunted' fish. An in a 'classically' stunted fish, is pretty much close enough to the reality for most people.
Think about a corrollary of the 'Organs do not continue growing' theory - that would be the 'fish grow to fit the size of the tank' theory. Neither is entirely true (the latter is an evil half-truth), neither is entirely untrue.
edit :^ all of this entirely my own surmisal and opinions of course, and not in the least bit backed up by any scientific evidence.