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.