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 $\mathscr S$.

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 $\mathscr S$.

The usage was apparently already well-established by 1956 when G. Mackey reviewed a French paper of F. Bruhat and wrote (first occurrence of the term in MathSciNet):

The representations concerned are assumed to yield “tempered representations” when restricted to the Abelian normal subgroup being studied. Here tempered means being not too badly unbounded in a precise sense suggested by Schwartz’s definition of tempered distribution.

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 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 $\mathscr S$.

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 $\mathscr S$.