Concerning
Have anyone ever introduced natural transformation in this "homotopical" way rather then the classical one in any reference like a textbook or some lecture notes?
Yes, Quillen introduces it in the paper
Higher Algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973.
In connection with his "Theorem A" and "Theorem B."