Can someone explain, why in English the name "tempered" wins?
Presumably because that’s how the inventor himself translated it (French past participle to English past participle), on e.g. p. 188 of Schwartz, Laurent, Mathematics for the physical sciences, Collection enseign. des sciences. ADIWES Internation Series in Mathematics. Paris: Hermann & Cie.; Reading, Mass. etc.: Addison-Wesley Publishing Company. 357 p. (1966). ZBL0151.34001:
A distribution $\mathrm T$ (that is to say a continuous linear form on $\mathscr D$) is termed a tempered distribution if it may be extended to a continuous linear form on $\mathscr S$.