Wikipedia says of stainless steel that stainless steel is "also known as inox steel or inox from French inoxydable".
As is well known, and mentioned in Wikipedia, stainless steel stains less easily and rusts or corrodes less easily, than ordinary steel, but it is not stain-proof and in some conditions can rust or corrode or give such appearance. For example, after a stainless steel surface is cleaned with steel wool rather than brass wool, it may develop rust stains.
Etymonline gives a 1917 date of origin of the term and says stainless steel is "so called because it is highly resistant to rust or tarnish." Edit: memidex.com tabulates definitions of stainless steel from a dozen sources, two etymology references, and several audio links. Wikipedia includes part of a 1915 New York Times article that refers to "a stainless steel" that is "claimed to be non-rusting, unstainable, and untarishable" [sic] which at least partly confutes the "stains less" etymology of stainless steel, which is suggested in the stainless-online.com link in the question.
With respect to American vs. British usage of stainless steel and inox steel, ngrams shows little difference between the corpuses, and in both cases inox has nearly no usage compared with stainless steel: