At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows.

Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{j_{1}\dots j_{q}}[\mathbf{f}] $$ to each basis $\mathbf{f}$ of an $n$-dimensional vector space such that, if we apply the change of basis $\mathbf{f}\mapsto \mathbf{f}\cdot R $ then the multidimensional array obeys the transformation law $$ T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p} T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}] R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} . $$

This is a standard definition I can remember reading in textbooks during my undergraduate degree. To me, it also seems far too confusing. To understand a $ (p, q) $-tensor as an element in $$ \text{Hom}(\underbrace{V^* \otimes\dots\otimes V^*}_{p\text{}} \otimes \underbrace{V \otimes\dots\otimes V}_{q \text{}}, \mathbb{K}) $$ one only has to understand the tensor product on vector spaces (which is easy to define in terms of bases). To then recover the description of a multidimensional array one also has understand cobases, however these can also be easily explained constructively.

Question

Why would anyone give the standard definition?

I initially thought the answer lay in applied mathematics. However linear maps are omnipresent in applied mathematics and I have never seen a linear map defined as a function on bases that satisfies coherence with respect to base change. Furthermore I feel the consensus would be that this is a bad definition from a pedagogical point of view (I certainly think it is). So why is the analogous definition of $ (p, q) $-tensors standard?