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To conclude with a note on terminology: having read carefully through the various comments to the OP and to this post, i think that it might be a good idea to speak of the present definition as the "transformation definition of a tensor", rather than the "standard definition", or the "physicist's definition" or the "pre-1930's mathematician's definition" or the "indices definition", just to collect a few of the terms that have been of use.

To conclude with a note on terminology: having read carefully through the various comments to the OP and to this post, i think that it might be sensible to speak of the present definition as the "phenomenological definition of a tensor", rather than the "standard definition", or the "physicist's definition" or the "pre-1930's mathematician's definition" or the "indices definition", just to collect a few of the terms that have been of use.

I think that the answer lies in the "educational culture" of physicists (as has already been implied in a couple of comments to the OP). Physicists are often used -well at least at the undergraduate level- to learn and perform complicated computations with abstract objects, without caring much about the structure and the abstract properties of the ambient spaces containing these abstract objects.
The definition of tensors as "generalized" vectors or matrices, with covariant and contravariant components, is one of such examples: Starting from such a definition, enables one to quickly learn how to perform computations with tensors, without demanding a deep understanding of the abstract definition of tensor products let alone dual spaces, manifolds, tangent and cotangent spaces or bundles etc. In this way, an undergraduate physicist quickly becomes able to perform computations in a wide range of topics (from classical mechanics to special and general relativity and from continuoum mechanics to electromagnetism and even field theories) while in most cases (s)he misses a deeper understanding of where all these objects "live".

I think that the answer lies in the "educational culture" of physicists. Physicists are often used -well at least at the undergraduate level- to learn and perform complicated computations with abstract objects, without caring much about the structure and the abstract properties of the ambient spaces containing these abstract objects.
The definition of tensors as "generalized" vectors or matrices, with covariant and contravariant components, is one of such examples: Starting from such a definition, enables one to quickly learn how to perform computations with tensors, without demanding a deep understanding of the abstract definition of tensor products let alone dual spaces, manifolds, tangent and cotangent spaces or bundles etc. In this way, an undergraduate physicist quickly becomes able to perform computations in a wide range of topics (from classical mechanics to special and general relativity and from continuoum mechanics to electromagnetism and even field theories) while in most cases (s)he misses a deeper understanding of where all these objects "live".

To conclude with a note on terminology: having read carefully through the various comments to the OP and to this post, i think that it might be a good idea to speak of the present definition as the "transformation definition of a tensor", rather than the "standard definition", or the "physicist's definition" or the "pre-1930's mathematician's definition" or the "indices definition", just to collect a few of the terms that are of use today.

To conclude with a note on terminology: having read carefully through the various comments to the OP and to this post, i think that it might be a good idea to speak of the present definition as the "transformation definition of a tensor", rather than the "standard definition", or the "physicist's definition" or the "pre-1930's mathematician's definition" or the "indices definition", just to collect a few of the terms that have been of use.