I'm not sure if I'm disagreeing with you, but I really dislike the "collection of numbers that transforms in this way" definition of vectors, tensors, etc. I get why it's useful in calculations (that's why physicists like it), but it seems really inelegant to me.
To me, a tensor is a function that maps a collection of m vectors to a collection of n vectors, such that every output is linear in each input. It's true that if you choose a basis, then you can write down an array of numbers which identifies the function, and that changing the basis causes those numbers to change in a certain way, but that's not what a tensor "is".
Of course, the beauty of mathematics is that there are many different ways of looking at the same object, so physicists can deal with their arrays of numbers, I can play with my multilinear functions, and everyone gets the same answer when we ask the same question.
I really dislike the "collection of numbers that transforms in this way" definition of vectors, tensors, etc.
It is not totally without merit if you consider the tensor bundle as associated to the principal bundle of linear frames. However, that's hardly suitable as an introduction to the topic.
To me, a tensor is a function that maps a collection of m vectors to a collection of n vectors, such that every output is linear in each input. It's true that if you choose a basis, then you can write down an array of numbers which identifies the function, and that changing the basis causes those numbers to change in a certain way, but that's not what a tensor "is".
Of course, the beauty of mathematics is that there are many different ways of looking at the same object, so physicists can deal with their arrays of numbers, I can play with my multilinear functions, and everyone gets the same answer when we ask the same question.