I don't agree that "tensor" is a good convenient term. E.g. C++'s std::vector does not really support vector operations, but most programmers can guess that it is a 1D array in terms of memory layout. "Tensor"? It's just a meaningless term to most beginners. If you think MultiDimensionalArray is too verbose, you may call it "MDA". It is shorter, conveys the meaning of the term better than "tensor" does and it's by no means misleading.

The only things I see to complain about are that vector spaces don't have to come with a choice of ordered finite basis and that arrays don't have to be of scalars from a field. But if you're already happy with the conflation of "vector" and "array", nothing new is happening in then speaking of multidimensional arrays as "tensors"; it's just continuing the same conflation.

Hmm, no, I am not happy with that. All I'm saying is that while std::vector is a bad name, most people can immediately get that it has an array-like memory layout. So, there is at least some merit with this inaccurate name. This is not the case with "tensor", a term that most programmers have never heard of (at least before the advent of TensorFlow).