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For the equivalence of definitions of quasiconformal maps the reference is J. Heinonen, Lectures on analysis on metric spaces, Springer 2001. Notice that the $K$ in the definiton you cite is not the same $K$ as in the Ahlfors definitions. So your definition of quasiconformality is equivalent to the usual one, but with a different $K$. Another reference is Lehto and Virtanen, Quasiconformal mappings in the plane, Springer 1973. The equivalence of all these definitions is a non-trivial fact.