In DPP, you do get the image with the picture style used for shooting as your first suggestion. Since it's all by Canon, they read each other's data and DPP knows which picture style was used. But when using RAW, you can then freely change the picture style after shooting, to get a different look, if you prefer.
Or you can do the same in some cameras, which allow RAW developing after the fact in the camera. If you have one of these (you have, since the 5D Mark III is a new camera), then you can apply different picture styles to the same RAW already in the camera, and check the result, if you like. But the computer screen is of course a better one to look at, so consider the camera processing option something being useful if you are for example travelling home from a photo session by bus or something.