$(\boldsymbol{1})\quad$ We call $\,P(n,k)\,$ k-permutations. Order matters, repetitions not allowed. The number of permutations of the n objects taken k at a time:
$$P(n,k)=n(n-1)...(n-k+1)=\frac{n!}{(n-k)!}$$
So, these are ordered arrangements/selections/choices. We also use $\;_n P_k\;$notation.
$\quad$
$(\boldsymbol{2})\quad$ When we permute all objects we simply call them permutations and write $\,n!$
$\quad$
$(\boldsymbol{3})\quad$ If repetitions are allowed and order matters, we refer to such arrangements as permutations with repetitions or distinguishable permutations:
$${{n}\choose{n_1, n_2, n_3…,n_p}} = \frac {n!}{n_1!\, n_2!\, n_3!…n_p!}$$
$$\quad$$
$(\boldsymbol{4})\quad$ When we have $n^k$ ordered arrangements, replacement allowed -- we call them permutations with replacements or k-tuples.
$$\quad$$
$(\boldsymbol{5})\quad$ When repetitions are not allowed and order doesn't matter, we call such arrangements combinations: ${{n}\choose{k}}\;$ or $\;_n C_k\;$ or $\;C(n,k).\;$ We choose n objects taken k at a time without regard to order. We read it "n choose k" and write:
$${{n}\choose{k}}=\frac{n(n-1)...(n-k+1)}{k!}=\frac{n!}{k!\,(n-k)!}$$
$(\boldsymbol{6})\quad$ And finally, when we deal with unordered arrangements, repetitions allowed -- we call them combinations with repetitions:
$${{n+k-1}\choose{k}}=\frac{(n+k-1)\cdot...\cdot n}{k!}=\frac{(n+k-1)!}{k!\,(n-1)!}$$
Please note it doesn't matter what these are called in German or French since each language has its own rules. An English manual of style is not applicable to other languages and vice versa; likewise you can't replace permutations or combinations with German variations in English. Yet we sometimes have different notations even within one language as authors may have their own preferences in terms of notation and terminology. There's nothing to worry about here.
Despite sounding a little pejorative to me, a German language speaker, it raises a question of whether such usage is really (internationally?) deprecated and considered outdated? -- No. That's more of a speculation on terminology used in other languages by some Wikipedia writers. See their editing history. Do not blindly trust something which is not a hard science in Wikipedia.
Do other languages possibly also name it differently? -- Yes. You can see it in the comments from people of other countries. There should be a lot of subtle examples. Let me make a rough guess based on "googling":
Disposizioni semplici=Variation ohne Wiederholung=l'arrangement=k-permutations
Arranjo com repetição=Variation mit Wiederholung=permutations with replacement.
So we should have been able to say these are just different technical terms -- but no -- it gets more complicated because in German terminology permutations are just a special kind of "variations". -- Yes, to some extent at least. Permutations is a broad term in English. What's more you can view these formulas from different "angles", e.g., combinations being just a special case of distinguishable permutations, permutations being just ordered combinations; and permutations with replacement can be called permutations with repetitions (it might create some confusion!) and so on.
The German flow chart with the German nomenclature? -- I checked it. It's really nice and logical. Quite commendable. I don't see how it may be inferior to any other nomenclature. It is rather on the contrary.
Nomenclatures, notations, and terminology differ from country to country. In biology, any species receives a binomial name (Latin name) and there's no ambiguity across the world about that species. In math we also have universal symbols and notations but they are not so rigid. You can find $cot^{-1}$, $arccot$, $arcctg $ used to denote the same and so forth. You can find in some countries analytic geometry is almost never part of calculus but always part of linear algebra. Sometimes you can find calculus being called mathematical analysis and being confused with analysis or real analysis. You may come across calques (or verbatim translations) of higher algebra, general algebra, etc. You may see how in English we coined words Calc I, II, III, IV as well as precalculus. Things are not clear cut, and there will be variations, just like the difference in meaning the word gift has in English and in German. While the word variation may have very similar meanings in English and German, there will also be differences, maybe subtle differences. And it is exactly the words with minor differences in meaning that cause most of the confusion. People expect them to be the same but they are not. One final example. We have books on vector calculus but it is a bit of a misnomer, as these books are just enhanced versions of Calculus III/IV. And it may have no bearing whatsoever on what might be the case in other languages.
CONCLUSION:
Now we can answer the "title" question: Why are permutations P(n,r) called variations in languages other than English? -- That is simply not the case! While European languages may use math variations in a similar way, the usage will diverge to some extent.
IMPORTANT:
Please note that not only technical terms are quite different from country to country but notations, too, may vary. Thus, in France, Russia, etc. permutations are often denoted $A^{k}_n$, and $C^{k}_n$ is used for combinations, which means the upper and bottom indexes are reversed. It may lead to mistakes in translation.
Best Answer
When we use quantifiers such as "$\forall$" and "$\exists$" we are quantifying over some domain. For example, the domain of discourse over which we are operating may be the natural numbers $\mathbb{N}$. Consider the following sentence: $$\forall x \exists y (x <y).$$ Given the context, this means "for every element $x$ of $\mathbb{N}$ there exists another element $y$ of $\mathbb{N}$ such that $x$ is less than $y$."
Now suppose we are not interested in all of $\mathbb{N}$, but rather every natural number less than $10$. Then we may use a bounded quantifier to express (the obviously false sentence) "for every number $x$ less than $10$ there is another number $y$ less than $10$ such that $x$ is less than $y$." Symbolically, $$\forall x <10 \exists y <10(x<y).$$ Since by using $x <10$ after the traditional quantifiers we are specifying that our numbers are coming from the set $\{1,2,...,10\}$ and not all of $\mathbb{N}$, the quantifiers "$\forall x <10$" and "$\exists y < 10$" are called bounded quantifiers.
Interestingly, most sentences using bounded quantifiers may be rephrased without using bounded quantifiers. For example, our sentence may be rewritten $$\forall x(x < 10 \to (\exists y(y <10 \wedge x < y))).$$