> 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 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.
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.