I recall multiple different definitions of holomorphic functions in several Complex Analysis textbooks, one by Ahlfors, one by Conway, and a couple more in Russian language textbooks.

One definition of a holomorphic function was based on Cauchy-Riemann equation, one starts from being locally analytic, one from angle preservating maps, and one from vanishing integral over closed curves.

And, of course, each textbook derived all other definitions as theorems.