I would like to enter the world of Derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute homotopy limits and colimits and others. There is also a relation with the theory of higher categories. History also says that Grothendieck, Heller, and in some sense Franke found the notion respectively for their own needs.

The references are varied regarding what axioms should one to impose on prederivators

Can you give a high explanation (motivation) for the use of the axioms of Derivators?

The background: category theory and some (not so serious) categorical homotopy theory