Hi, I was wondering, if there is a way to make a copy of a RAW file from the processing settings that the camera intially applies upon taking a RAW photo.
No...what you are seeing is a JPEG preview rendered by the camera. Camera Raw doesn't use the camera maker's software so there is no way to have Camera Raw any settings other than white balance.
Camera Raw creates its own default rendering based on Camera Raw's processing routines. So you'll need to make the adjustments using Camera Raw's toolset to render the image the way you want.
Also note there is nothing particularly "correct" about the camera's rendering...it's simply a look designed by the camera maker which is generally more saturated and higher contrast–both adjustments Camera Raw can do equally well.
Camera Raw also offers many vender matching DNG Profiles in the Camera Calibration panel that can change the way colors are rendered...you may want to look at those.
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There is software that will display and save the JPEG image that's embedded in the raw file.
For example, IrfanView can be set to extract the JPEG from a Canon .cr2 file and display it. It's a simple File - Save As away from saving it as whatever you like.
I believe the Canon software that comes with the camera can extract the JPEGs directly and save them (or at least it could at one time, I haven't used it in an age).
P.S., it is also possible to set up defaults in Camera Raw to match the in-camera JPEGs very closely. While Jeff may not like the look his camera gives his JPEG files some of us do, and I find it makes a good starting point, as there's just one "image" to start with, no matter how you view it.
The other option is to use Nikon NX2 or the free ViewNX2 to output JPG or TIF files using the in camera settings.