This seems excessively nit picky. Is the 'heat treatment' technically part of the forging? Technically no but for the layman's idea of forging it probably is ok to say yes.
Did that one line in the article perpetuate the trope? In the sense that it means 'merely wielding a katana makes one a superhuman fighter' I would say 'clearly no' but in the sense of 'katana is the coolest sword' perhaps. Did katanas have world renowned sharpness? I would say yes. World renowned 'strongness' (which can have a lot more meanings)? Notsomuch.
Its basically the Katana folds, except you can explicitly see the folds between multiple kinds of steel. Damascus Steel is also the stuff of legends, because it is literally a technique lost to history (although people claimed to have reverse engineered it).
Furthermore, some evidence points that true Damascus Steel had carbon nanotubes and cementite nanowires strengthening the blade (possibly created through the unknown ancient forging process). Not bad for Steel made in the BC eras.
The katana making techniques (pattern welding, differential HT) were not unique. So is the performance. Claiming otherwise is perpetuating the myth. The Japanese sword got its "world renown sharpness" the same way magnum .44 become the "world's most powerful handgun" after Dirty Harry. Also swords are not very sharp even with highly polished edge. Too high angle. If you want to see something really sharp - take a walk to some of the better restaurants.
I disagree, forging has a very straightforward meaning, heating the metal to red-hot, and hammer away to form it. Items are typically then quenched in oil or water, which cools steels quickly enough that they are quite brittle. Heat-treating them then tempers that, allowing the smith to achieve whatever level of hardness (with brittleness) or flexibility (with loss of hardness and thus sharpness). The Japanese smiths do seem to have gone to great lengths to forge-weld steels of differing hardnesses together in making blades - I was amazed to learn that the curve of the katana results from quenching a blade that is forged basically straight, but has much harder steel along the edge (which can't shrink as much as the more or less iron of the spine, due to carbides).
Forging does not create a sharp edge. The micro structure that gives sharp edge is created during the heat treatment.