According to the very wikipedia article cited in the original question:
The story of the salting of Carthage was probably invented in the 19th century.
(In the section quoted in the question:) The concept of salting destroyed cities originated in the ancient middle east and was revived in late medieval Europe. Specifically the Hittites and Assyrians used to ceremonially spread minerals and weeds across destroyed cities. Take an example, Ashurbanipal quoted by wikipedia
(...) I devastated the provinces of Elam. Salt and sihlu I scattered over them (...) The dust of Susa, Madaktu, Haltemash and the rest of the cities I gathered together and took to Assyria (...) The noise of people, the tread of cattle and sheep, the glad shouts of rejoicing, I banished from its fields. Wild asses, gazelles and all kinds of beasts of the plain I caused to lie down among them, as if at home.
Sihlu is some plant (of the thorny kind), Susa, Madaktu, Haltemash are elamite cities. So he says, he salted not just the cities but the entire country. What he actually wants to convey is that it is now pretty desolate; no happy farmers, no domestic animals, but weeds and wild beasts.
I am, therefore, not convinced that it is meant literally, both in the sense that perhaps they actually scattered nothing at all (except for what accidentally fell over while they were looting) and in the sense that if they did, it was probably not valuable table salt but anything toxic they could find. (And, as has been pointed out, it would clearly have been ceremonially.)
Note that much of the farm land of the middle east had to be irrigated artificially and was prone to salinization. So "scattering salt" may also actually refer to driving away the farmers that would tend to the land.
With regard to the revival of the custom in medieval Europe (wikipedia cites Pope Boniface VIII), I would assume they just had red it in the bible (Judges 9.45) and thought it particularly godly to revive the old customs. It would still have been ceremonially. Though historical evidence suggests that they, perhaps different from Ashurbanipal, actually performed the ceremony. That having become a common cultural institution, it may in the 19th century have been projected back into Roman Antiquity.