Addressing just the primary question, I think there are clear examples
where English-speaking mathematicians have been slow to catch up with
developments well-known to German or Russian speakers. One I would
mention is the result that surface mappings are generated by twists,
published by Dehn in 1938, but rediscovered by Lickorish in 1963. The
big advances in the theory of surface mappings due to Thurston in the
1970s were, I think, somewhat slowed by the fact that he had to
rediscover many results of Nielsen published in German or Danish in
the 1920s.
At a more trifling level, I was once embarrassed to learn that a result
I published in 1987 was well-known to Russian mathematicians.
When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theorists that analysts (and others doing what you call realistic mathematics) would adopt ZF + DC + "all sets of reals are Lebesgue measurable" as their preferred foundational framework. There would be no more worries about "pathological" phenomena (like the Banach-Tarski paradox), no more tedious verification that some function is measurable in order to apply Fubini's theorem, and no more of various other headaches. But that expectation wasn't realized at all; analysts still work in ZFC. Why? I don't know, but I can imagine three reasons.
First, the axiom of choice is clearly true for the (nowadays) intended meaning of "set". Solovay's model consists of certain "definable" sets. Although there's considerable flexibility in this sort of definability (e.g., any countable sequence of ordinal numbers can be used as a parameter in such a definition), it's still not quite so natural as the general notion of "arbitrary set." So by adopting the new framework, people would be committing themselves to a limited notion of set, and that might well produce some discomfort.
Second, it's important that Solovay's theory, though it doesn't include the full axiom of choice, does include the axiom of dependent choice (DC). Much of (non-pathological) analysis relies on DC or at least on the (weaker) axiom of countable choice. (For example, countable additivity of Lebesgue measure is not provable in ZF alone.) So to work in Solovay's theory, one would have to keep in mind the distinction between "good" uses of choice (countable choice or DC) and "bad" uses (of the sort involved in the construction of Vitali sets or the Banach-Tarski paradox). The distinction is quite clear to set-theorists
but analysts might not want to get near such subtleties.
Third, in ZF + DC + "all sets of reals are Lebesgue measurable," one lacks some theorems that analysts like, for example Tychonoff's theorem (even for compact Hausdorff spaces, where it's weaker than full choice). I suspect (though I haven't actually studied this) that the particular uses of Tychonoff's theorem needed in "realistic mathematics" may well be provable in ZF + DC + "all sets of reals are Lebesgue measurable" (or even in just ZF + DC). But again, analysts may feel uncomfortable with the need to distinguish the "available" cases of Tychonoff's theorem from the more general cases.
The bottom line here seems to be that there's a reasonable way to do realistic mathematics without the axiom of choice, but adopting it would require some work, and people have generally not been willing to do that work.
Best Answer
Many topics in linear algebra suffer from the issue in the question. For example:
In linear algebra, one often sees the determinant of a matrix defined by some ungodly formula, often even with special diagrams and mnemonics given for how to compute it in the 3x3 case, say.
det(A) = some horrible mess of a formula
Even relatively sophisticated people will insist that det(A) is the sum over permutations, etc. with a sign for the parity, etc. Students trapped in this way of thinking do not understand the determinant.
The right definition is that det(A) is the volume of the image of the unit cube after applying the transformation determined by A. From this alone, everything follows. One sees immediately the importance of det(A)=0, the reason why elementary operations have the corresponding determinant, why diagonal and triangular matrices have their determinants.
Even matrix multiplication, if defined by the usual formula, seems arbitrary and even crazy, without some background understanding of why the definition is that way.
The larger point here is that although the question asked about having a single wrong definition, really the problem is that a limiting perspective can infect one's entire approach to a subject. Theorems, questions, exercises, examples as well as definitions can be coming from an incorrect view of a subject!
Too often, (undergraduate) linear algebra is taught as a subject about static objects---matrices sitting there, having complicated formulas associated with them and complex procedures carried out with the, often for no immediately discernible reason. From this perspective, many matrix rules seem completely arbitrary.
The right way to teach and to understand linear algebra is as a fully dynamic subject. The purpose is to understand transformations of space. It is exciting! We want to stretch space, skew it, reflect it, rotate it around. How can we represent these transformations? If they are linear, then we are led to consider the action on unit basis vectors, so we are led naturally to matrices. Multiplying matrices should mean composing the transformations, and from this one derives the multiplication rules. All the usual topics in elementary linear algebra have deep connection with essentially geometric concepts connected with the corresponding transformations.