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In the comments I offered the "classical" example of Grothendieck's definition of a scheme. But for a more modern example: Fomin and Zelevinsky's definition of a cluster algebra is at first sight a rather ungainly thing, but turned out to be exactly what was needed to unify certain patterns that appeared in various hitherto unrelated areas of mathematics like discrete integrable systems, the representation theory of associative algebras, Poisson/symplectic geometry, etc.