I'm reading a book on Geometry from the '70s and when speaking about Tensors it defines them starting from the covariant and contravariant commutation rule. I know this definition was quite widespread at least until the '90s.
I was used to approach Tensors as multilinear applications, but now I'm wondering: was this definition of the covariant/contravariant transformation rule really bad? Does it have any effective flaw or is it more a question of taste choosing between the two definitions?
I have to choose a presentation for tensors in a Differential Geometry environment and in a really selfcontained and simple way, so I'm wondering if the old way with transformation rules is an option or not
In my opinion this is in fact mostly a question of taste, but not completely. Different approaches have different advantages. Coordinate system based approaches (those with all the indices, this is what you are referring to when you are referring to the transformation rules. These are just the statement of coordinate invariance of the geometrical content of the equations) are often useful for partial differential equations (especially elliptic second order equations in my personal experience), since it may be easier to apply exisiting results from PDE theory.
Other approaches allow, for example, easy and efficient manipulation of equations but may make it difficult to determine the geometrical content of an equation (this is often true for differential form base approaches, which admit application of powerful algebraic methods), while the geometric meaning may be easier to discover in an invariant vector field base approach (because this is often closer to our intuition).
In all cases your mileage may vary.
The advantage which is of highest importance to you is usually the best choice for you. This may result in the choice of different formalisms depending on the problem you are looking at.
For the beginner the choice is likely to be mostly influenced by how you have been introduced to the topic. If you start to work seriously on the topic you will have to read papers in which different approaches are used and, that way may have to learn other approaches.