Catching up on chapter 3

Reading through chapter 3 we come across this:

which is not a properly defined recursive function.  The text then goes on to say

"Superficially, the reason appears to be that it defines f'(n) in terms of the value of f' at a larger argument, f'(n+1). It is true that it does not define a function, but our reasoning as to why that is so is not adequate."

and then goes on to show how it is equivalent to this

in efforts to somehow dispel our previous faulty reasoning.
I'm struggling to see how writing the function in this latter sense with a =>  in the place of a > somehow mediates the concern that it's defined in terms of higher and not lower terms... because it still is defined in terms of higher terms.  Maybe it has to do with no proper base case?  I'm not sure I get it. I'm going to as the T.A.s