It seems to me that the people I know who are effective influencers are also people who are willing to own their influence. They don't pretend that they are disinterested. Sometimes they don't even pretend that they are particularly interested in you or your well being. They own that one thing that they want to change in you.
I recently read a comment in a Linked In group by a coach who believed that he could not change anyone's perceptions, he could only open up possibilities. This is nonsense. You cannot change anyone else's conscious choices but it is a relatively easy thing to change their perceptions and thereby to influence the choices they make unconsciously. If you are not sure that this is true, stand on a busy corner and point up at a building as though you were seeing something really interesting. Then watch what happens. You will find that some people stop walking, change their posture, and become curious. You have effectively changed their perspective.
It is inevitable that we will change each other's perspectives over and over again. The human brain has many functions specifically designed to notice other human beings. This leads us, consciously or automatically (unconsciously) to change what we experience because we are influenced by what other people are noticing or experiencing. This is how we learn from each other, how we predict who will help us and who will hurt us, and how we influence the thoughts, choices and actions of other people.
If you're in a bad mood, you are changing other people's perceptions. If you are obsessed with a problem, you are changing other people's perceptions. If you have to do it your way (the 'right' way), then you are changing other people's perceptions. You don't have to engage with them. You just have to be within the range of their senses. Watch how a yawn travels around a public space where no one is particularly paying attention to anyone else. Now consider that your state of mind is as much an influence in a room as the temperature or the lighting.
Own it. Understand that the way you are is more influential than what you say (although what you say can also matter). Understand that people are changing all the time, and some of the time it is in relation to you. You have an opportunity to make the people around you happier or more driven or more balanced. That opportunity begins with your willingness to experience the change you want to influence and to actively intend to have the influence you want.