The goal of influence is often manipulation... we see it all the time with "influencers" trying to illegally advertise to people by not disclosing sponsorships
People influencing others are trying to manipulate their emotions or thoughts into feeling a certain way about something...
We can look to the definitions themselves...
influence: the capacity to have an effect on the character, development, or behavior of someone or something, or the effect itself.
manipulation: control or influence (a person or situation) cleverly, unfairly, or unscrupulously.
influence is literally there in the definition of manipulation
I think you might want to re-read all of this again.
Manipulation is non-consensual and controlling influence, yes. That part is important.
Just like the difference between a house and a jail is essentially if you are being forced to be there against your will or not.
They both are structures people live in and that provide a degree of structure and protection, but one is generally considered okay or even good, and the other is considered pretty bad and terrible. Largely because one you can choose or do things to change, the other you get stuck in against your will and the environment is outside your control.
Most ‘influencers’ also cross pretty strongly into manipulation many times, so using the title is about as accurate as ‘Democratic Peoples Republic of z’.