This may not be an entirely satisfactory answer, in part due to the paucity of primary source material, but also due to the fact that modern research into early Hungarian coaches seems - unsurprisingly - to have been published almost exclusively in Hungarian.
The primary innovation of the Kocs coach-builders seems to have been a frame built from flexible hardwood, which offered a much more comfortable ride to passengers than other contemporary designs.
There appears to be no contemporary evidence suggesting that these Kosci used steel-spring suspension.
As far as steel-spring suspension is concerned, it is worth noting that the Hungarian Wikipedia page for Kocsi makes no such claim. Indeed, the English Wikipedia page that you quote also notes that:
A 16th-century German depiction of a kocsi without springs puts this theory in doubt, however, and it is uncertain whether the springs or some other feature were responsible for the spread of the word throughout Europe.
The earliest description of the Kocsi that I could find comes from the autobiography of the diplomat, writer, and historian, Sigismund von Herberstein. A modern copy of this was published by Kovachich in 1805. This is, in turn, quoted by several authors.
He travelled in one of these Kosci in 1518, and wrote that:
"They are named after a village ten miles from Buda (Kocs); they are drawn by three horses running abreast of each other, when there is little or no ice on the ground. They carry four people along with the driver, and is a very comfortable conveyance".
The vehicles appear to have had the following features:
A wooden frame with 'saddle-shaped' sides made of wickerwork.
Passengers sit on raised seats at the back above the larger back wheel.
The carriage is open.
It has an elastic hardwood structure with almost no ironwork.
This description appears to correspond with the earliest known depiction of a Hungarian Kosci, published by Jeremias Schemel of Augsburg in 1568:
It seems to have been the works of Sigismund von Herberstein and Jeremias Schemel that made the Hungarian Kosci popular across Europe.
Kemecsi Lajos published some research into the characteristics of a modern reconstruction of a Hungarian Kosci is his paper A mezıkövesdi Tájházi Mőhely elıadásaiból - A magyar kocsi (From the performances of the Fieldworks Workshop in Mesőkövesd - A Hungarian Kocsi).
This is one of the modern sources that quotes from Sigismund von Herberstein (and one of the few that I could find online that aren't behind paywalls!) and includes a diagram of their reconstruction of the vehicle: