A new definition of recursively enumerable set?
Your definition makes no sense because according to it every language is recursively enumerable. Given any $C \subseteq \Sigma^*$, take $B = C$ and $f(x) = x$, the identity function, which is computable. Then $f^{-1}(C) = C$, therefore $C$ is recursively enumerable.

We can fix your definition by observing that the following theorem holds:


  Theorem: A language $A$ is recursively enumerable if, and only if, there is a partial computable $f$ and a recursive set $B$ such that $A = f^{-1}(B)$.


You are close to reinventing the notion of many-one reduction, by the way.