As I understand it (someone please correct me if I'm wrong), the picture style you have set, is used to make the thumbnail/jpg that is shown on the back screen, so set it to what you want and use that to show people on camera....but as other said, it has nothing to do with the RAW that is output out the camera (if shooting RAW).
I have a couple of the cine styles on my camera in the 3 user defined styles. I click to those when shooting video, and occasionally, if I want a nicer pic on the back like you described, when shooting stills, I switch to one of the Canon included styles...normal or neutral usually.
Lately, since I don't show the back to people much, I just leave it on my marvels cine style all the time that way I don't accidentally end up shooting video and forget to switch to it first.
But if you don't mind remembering to shoot back and forth, just do that between your video and stills...just a couple of dial clicks to change back and forth.