sorry I edited my comment because I wasn't quite sure what I meant by that either. I am too sleepy.

But, I disagree. partial functions B->A are just a superset of functions B->A, so requiring g to be partial is just relaxing a requirement. Whereas requiring g: im(A)->A adds an additional restriction, potentially breaking any place that concept of a left inverse is used. Invariably someone will just define a new thing equivalent to what is normally called the left inverse because they still need the concept for some reason.