Could you explain what are the differences in these two and which one should be used for files to be sent to press?
I wasn't able to find any good answer to this question.
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Document Color is the color mode that your document will be output to when exported for print.
Working CMYK is the CMYK values you are using within your working document. So if your document were RGB, but you wanted to make a color based off of CMYK color values, they would be working CMYK values.
So a CMYK document might look normal on your screen but weird on the web, and vice versa. That kind of thing.
Both should be used for press. Ideally you should always match the working document with the same color mode. CMYK with CMYK, RGB with RGB. The other answer explained it well in a simple way; one way or another your file is going to be output either by exporting into another file format or using a "save as" or printing directly from the file.
The color space of the file itself is the color values being used in each files. Often used for Photoshop or Illustrator files, and also values of your swatches.
The "working document" is how your colors are being interpreted and shown to you by the software. Often used in the "main publishing document" where you assemble all your images together, like InDesign or Illustrator.
So if you're working with a CMYK image on a RGB working space, your CMYK colors will be changed to RGB values. For example, you import an image with a black K100 (a CMYK pure black) in a RGB document, that black will be converted to something like C55-M76-Y24-K78 instead of C0-M0-Y0-K100.
That's why you should simply use the working space in CMYK and the matching CMYK color mode or values when working on print projects. It's not a good habit to "convert" your colors on export, especially when you convert from RGB to CMYK. You lose way too much control of the CMYK values of your colors and can end up with weird results or rich blacks where you shouldn't have them.
Same goes for RGB, you want to reach the maximum colors in that gamut; you'll use RGB color mode in a RGB working document.