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 uses 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."

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."