> Mathematicians' tensors = generalization of arrays, transformation rules don't matter.

That's definitely inaccurate, at least it doesn't match what I think of as tensors in mathematics.

In mathematics, tensors are the most general result of a bilinear opteration. This does imply that they transform according to certain laws: if you represent the tensor using some particular basis, that basis can be expressed in the original vector spaces you multiplied, and choosing a different basis for your vector spaces results in a different basis for your tensors.

By "most general bilinear operation" I am talking about what is expressed in category theory as a universal property... with a morphism that preserves bilinear maps.

Tensors can be over multiple vector spaces or a single vector space (in which case it's typically implied that it's over the vector space and its dual). When you use a vector space and its dual, I believe you get the kind of tensor that physicists deal with, and all of the same properties. Note that while vector spaces and their dual may seem to be equivalent at first glance (and they are isomorphic in finite-dimensional cases), both mathematicians and physicists must know that they have different structure and transform differently.

Something that will throw you off is that mathematicians often like to use category theory and "point free" reasoning where you talk about vector spaces and tensor products in terms of things like objects and morphisms, and often avoid talking about actual vectors and tensors. Physicists talk about tensors using much more concrete terms and specify coordinate systems for them. It can require some insight in order to figure out that mathematicians and physicists are actually talking about the same thing, and figure out how to translate what a physicist says about a tensor to what a mathematician says.

But they are not totally different! The Physicists' tensors are actually Mathematicians' tensors, but parametrized by some parameters (coordinates). Then you have special laws what happens if you make a (possibly non-linear) change of the coordinates. See https://en.wikipedia.org/wiki/Tensor#Tensor_fields