Influencing is not manipulation.
Manipulation has a hidden agenda and is about tricking someone into doing what they don't want to do.
Influencing is getting folks to feel safe/open enough to take in new/different information and consider other courses of action.
I have manipulated a three-year old into going to bed (haven't we all?), proudly turning off the light at the end. I have tricked someone into showing up for a surprise party they didn't know we were throwing. I have been deliberately mislead into signing a document for a new water heater I did not need (somedays I am not smart). Not all manipulation is for evil intent. However, the common themes are misleading or misrepresenting information to get someone to do something they might not otherwise choose.
You try to keep control of the other party when you manipulate.You can only manipulate most folks once. It's a one-time deal and builds no trust, confidence or relationship.
Influencing is the other side of the coin. It's about relinquishing control and offering advice or guidance (which may or may not be taken). Influencing is about committing to offering an open, honest dialogue, taking responsibility for offering the best information one has but not directing the application of the information or dictating the choices.
Mentoring is influencing at its best.