The way I use the term, manipulation is influence that is improper either due to the methods used, or the social context in which they are being used. Our society (and I suspect any complex society) will have very complicated norms about what kinds of influence are legitimate in what circumstances. For instance, lying to induce somebody to do something they wouldn't do if you told them the truth is improper manipulation in most circumstances.
A lot of our norms are area-specific. For instance, in law, there's a very well developed set of principles about precisely what constitutes misrepresentation or undue influence -- in which case the contract can be void.
I suspect, in general, there will not be good reasons for putting the demarcation at one point over another: I imagine that social processes will drive norms towards at least a local cost-benefit maximum. And if the cost-benefit landscape is smooth, those local maxima will be places where there landscape is locally flat, implying that there's negligible harm if you move the equilibrium slightly.
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.