Showing that the number of primitive-recursion programs for each function is countably-infinite
I proved that there are infinite number of definitions using the following construction


That works in principle, but be careful about what $f$ is in that term, formally.


  Now the issue is that this is not the only way to construct an infinite number of equivalent definitions


Right you are!


  How can I show that there is a 1−1 correspondence of the set of all primitive recursive definitions with the set of naturals numbers?


You don't have to do this explicitly.
Use that the class of countable sets is closed against taking subsets.