Yes, very simple, except that when physicists say "tensor", they mean tensor fields, on smooth, curved manifolds, in at least four dimensions, often with a Lorentz metric. Things stop being simple quickly.
A tensor field is not a tensor, but the value of a tensor field at any point is a tensor, which satisfies the definition given above, exactly like the value of a vector field at any point is a vector.
The "fields" are just functions.
There are physics books that do not give the easier to understand definition given above, but they give an equivalent, but more obscure, definition of a tensor, by giving the transformation rules for its contravariant components and for its covariant components at a change of the reference system.
The word "tensor" with the current meaning has been used for the first time by Einstein and he has not given any explanation for this word choice. The theory of tensors that Einstein has learned had not used the word "tensor".
Before Einstein, the word "tensor" (coined by Hamilton) was used in physics with the meaning of "symmetrical matrix", because the geometric (affine) transformation of a body that is determined by the multiplication with a symmetric matrix extends (or compresses) the body towards certain directions (the axes that correspond to a rotation that would diagonalize the symmetric matrix). The word "tensor" in the old sense was applied only to what is called now "symmetric tensor of the second order" (which remains the most important kind of the tensors that are neither vectors nor scalars).
I think this is far too simplistic, for one because the values of this putative function depend on the chosen coordinate system.
So I completely agree with the comment you are replying to: when a physicist says "tensor" they really mean a "tensor field" and the definition of the latter is quite a bit more involved than just specifying a multilinear map at each point of a manifold.
This is the essence of notions like scalar, vector, tensor, that they do not depend on the chosen coordinate system.
Only their numeric representations associated with a chosen coordinate system do depend on that system.
If you compute some arbitrary functions of the numeric components of a tensor in a certain coordinate system, in most cases the array of numbers that composes the result will not be a tensor, precisely because the result will really be different in any other coordinate system, while a tensor must be invariant.
All physical laws are formulated only using various kinds of tensors, including vectors and scalars, precisely because they must be invariant at the choice of the coordinate system.
I insist that calling them "just functions" is simplistic. In fact, I'd say that the complexity of your elaborations kind of proves my point.
Note that I deliberately use "simplistic" and not "wrong", since a section is a function of sorts.
Just as a (p,q) tensor is a multilinear object related to a single vector space, a tensor field is a section of a tensor bundle associated to the vector bundle. (A section is just a function on the underlying space whose value at a point lies in the vector space above the point.)
Usually, the vector bundle relevant in physics is the tangent bundle of a 4-manifold.
This abstract way of defining tensors and tensor fields is manifestly invariant under coordinate changes, but it takes some machinery to set up. Whereas the 'numbers associated to each coordinate system which transform in a certain way' is more direct, but the rules can seem arbitrary at first sight. Also, maybe this approach can generalize to allow more transformation rules which might take some time to put into an abstract setting.
Standard example is a matrix A which transforms as PAP^-1 (where P is linear coordinate change) vs a matrix T which is a linear map between vector spaces.
The same issue appears in software where you can expose a data structure as a tuple of numbers/string fields and then define functions on them, or you can expose it as an abstract data type where the user of the library can only apply certain functions on them and the implementation author can choose different representations(coordinate changes) in which to easily compute the functions.
The definition of a tensor as linear maps, while simple to understand, has no content that is useful for doing physics. To do any physics, or for that matter, any geometry with tensors, you need to define the notion of covariance and contravariance.
Besides, the starting with the latter notions allow you to define tensors more naturally. You start with trying to understand how geometric objects transform under coordinate transformations, and you slowly but surely end up with tensors.
All the physical quantities that are defined to be tensors are quantities used to transform either vectors into other vectors or tensors of higher orders into other tensors of higher orders (for instance the transformation between the electric field vector and the electric polarization vector).
Therefore all such physical quantities are used to describe multilinear functions, either in linear anisotropic media, or in non-linear anisotropic media, but in the latter case they are applicable only to relations between small differences, where linear approximations may be used.
The multilinear function is the physical concept that is independent of the coordinate system. The concrete computations with a tensor a.k.a. multilinear function may need the computation of contravariant and/or covariant components in a particular coordinate system and the use of their transformation rules. On the other hand, the abstract formulation of the physical laws does not need such details, but only the high-level definitions using multi-linear functions, and it is independent of any choice for the coordinate system.
There is a unique multilinear function a.k.a. tensor, but it can be expressed by an infinity of different arrays of numbers, corresponding to various combinations of contravariant or covariant components, in various coordinate systems. Their transformation rules can be determined by the condition that they must represent the same function. In the books that do not explain this, the rules appear to be magic and they do not allow an understanding of why the rules are these and not others.
Obviously these things are not just useful to physics, but are indispensable, and so I think the assertion that only the definition of tensor that is useful to physics is the definition tensor=multilinear map is somewhat out of step. Perhaps it would be better to assert that the concept of multilinear map is essential to every useful definition of tensors in physics.
This is exactly the point. Abstract physical laws must be invariant to coordinate transformation. From a pedagogical point of view, perhaps this is less important when discussing anisotropic media, but critical when discussing general relativity. Hence, the first reason why many physicist book writer think it very important that covariance/contravariance of tensors be central to both their definition and their pedagogy as applied to physics. You have to convince the student that tensors are the right mathematical objects to describe reality because they preserve this invariance.
The second reason is just as important. Physics is nothing without validating abstract physical laws by experiment. And that validation can not be done without computing predictions. Which in turn will require the right coordinate system, which will require covariance/contravariance of tensors. You can't just disregard these computations as unimportant or unnecessary from either a pedagogical point of view or a deeper philosophical one.
Why always think of tensors as "functions"? In physics, we often think of them as "quantities" - scalar, vector, etc.