Influence and manipulation are both attempts to alter the actions or propensity to act of another agent. The only difference between the two that can generally relied upon is that the one who calls it manipulation finds it distasteful or immoral. If you disagree I invite you to find a general principle cleanly dividing the following examples into manipulation and influence; all the better if it can be made uncontroversial.
A woman telling her long-term boyfriend that they're not getting married before he gets his doctorate.
A man teasing a friend.
A man teasing a female friend.
A man teasing a female friend, flirting.
A man teasing a female friend, flirting with intent.
A man teasing a woman he met ten minutes ago, flirting with intent.
A woman encouraging her son to become a teacher because the job security is good.
A woman encouraging her son to become a lawyer because he'll be better able to support her in her old age.
I'm not denying that manipulation and influence can be usefully distinguished. I do not believe they can but I haven't spent the last month thinking about it on and off. There might exist a Schelling point dividing bidirectional communication into socially acceptable influence and unacceptable manipulation, a lawyerly thing, useful but without any defensible reason to be there exactly but that it must be somewhere and we have come to an agreement that here will do. If you believe that there is a point that actually is pragmatically better than another by a real margin, please state it, and defend your proposition.