"I influenced the outcome" and "I partly controlled the outcome" strike me as synonymous in typical contexts.
To be frank, I find the notion that it is inherently unimmersive to play a PC who non-magically imposes his/her will on others strikes me as bizarre. For instance, when a police officer, bouncer or security guard looks at you and signals that something is not to be done, or a certain place not to be entered, s/he imposes his/her will on you. There would be nothing inherently un-immersive about building such a PC with a "none may pass" power.
Am I really the only poster on this forum who has experienced this phenomenon in real life?
Cops would love to have such a uniformly controlling ability. They don't have it.
What a cop has is a degree of societal respect backup up by the known consequences of not heeding his direction. In effect, police officers have a form of Intimidate -- "Do this or else" backed up by a large list of "or elses" they can call upon. It is up to the individual to determine how seriously to take the intimidation and to determine what behaviour to adopt in its presence. People ignore cops, do end runs around them, take swings at them, run from them, etc. All. The. Time. In other words, people continue to have agency.
A police officer cannot guarantee the reaction of anyone with whom they interact. That's why their trained to handle themselves for when it doesn't go the way they expect/prefer.
Authorial powers remove agency or rewrite the universe to create a justification for new behaviours that have no requirement to make previous actions or future actions by the character cosistent or plausible.
And we're not talking just CaGI. I have similar problems with the Warden -- what if I purposely put my character in a particular location for sound tactical/roleplaying reasons and the Warden ally shifts it away? Even if there are strong reasons to stand /here/ and not /there/?