I think we usually don't need to compare these two. But if we compare them we would find both of their significance. The first one doesn't consider each individual specifically; I feel. When we put the first one into practice there is a great chance of dilemma in many occasions. There is a chance of ignoring that particular sinner's individuality/mind. Most often the person who is treating a sinner may ignore the quality 'love'. I mean, loving his 'essence'(soul)...(Even if he didn't forget one or two qualities for humanity.)
Often, circumstances make a sinner. If you hate the sinner always, how could he become a good person? Just remember the (author's) background of the Ramayana--Story of Ratnakara (Valmiki). Would we ever get that great Epic if he were ignored? How did Rama treat Ravana always (in the Ramyana)? What was his attitude towards him?
So, I would say that the first one pleases most people, but the second one pleases the wise as well as the sinner and gives a motivation for him to become good. Certainly, you can please most people with the first statement. So you may give the first one as an advice to (almost) anybody; but not the second one. (Suppose) When dealing a sinner sometimes you would also become a 'sinner' accidentally. How should we treat you then? Just remember that.
If you wish to take the essence of both, I would say: The usage 'ove for mankind' made the first statement 'milk' and the usage 'love the sinner' made the second 'butter'.
I'm just asking whether any philosopher has pointed that out, or made
a similar claim about the particularization of similar ethical maxims.
I don't know whether anybody had ever tried to discriminate the individualization of Gandhiji's words.
So, to your question,
Does Gandhi's paraphrase of Augustine's phrase distort its meaning?
If churning is any kind of distortion or deformation, Gandhiji's statement is also a distortion; otherwise not. This might mislead the people who can't assimilate the meaning in its correct sense. Many people would misunderstand you and your idea if you use these words of Gandhiji for giving advice.