>Can someone explain, why in English the name "tempered" wins?

Presumably because that’s how the inventor himself translated it, e.g. on p. 188 of 
<cite authors="Schwartz, Laurent">_Schwartz, Laurent_, Mathematics for the physical sciences, Collection enseign. des sciences. ADIWES Internation Series in Mathematics. Paris: Hermann &amp; Cie.; Reading, Mass. etc.: Addison-Wesley Publishing Company. 357 p. (1966). [ZBL0151.34001](https://zbmath.org/?q=an:0151.34001):</cite>

>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 $\mathscr S$.