I'm sure David will have a response, but here is my two cents worth (along with a money-back guarantee:) ...

I don't know the technical definition of influence, but when I use that term, it means to motivate to action.  "Coerce" seems like a better choice for the type of motivation provided by weapons or rules.    It is certainly involuntary, again as opposed to how I normally think of "influence".

"Influence" just seems a kinder and gentler word ...

John