Make sure you follow the trouble shooting steps above as this is undoubtedly a bad monitor profile or a gpu driver issue but wanted to comment on this:
>I've researched the problem and it seems that the Develop module and the Library module use different color profiles? - Unless I'm missing something, how is that a good idea? Why would your own platform use two different color settings between tabs?
there is a very simple reason for this difference. The non-develop modules all render from jpeg previews. The develop module renders directly from raw in the widest color space available. It uses a color space that has prophotoRGB primaries but linear gamma (since raw is a linear capture). When converting to a jpeg preview, you cannot use that colorspace as you always need to employ a gamma curve in jpeg. Also jpegs in prophotoRGB space tend to show banding, so Lightroom uses a smaller color space to avoid this banding. It is likely that the red color you dialed in is way outside of standard color spaces such as sRGB, AdobeRGB or displayP3 and that causes the color shift you observe. This color shift will also occur therefore if you export your images for sharing. Instagram for example is sRGB only and there is no way it can display this red. In the develop module you can soft proof your images for these standard color spaces. Try turning it on and use sRGB for the proofing space. You will see the same shift. This is simply a limitation of the color spaces that you might be exporting in. lastly if you are editing in HDR, which allows far brighter saturated colors, be aware that you need to enable hdr previews in library. Also, be aware that it is not super likely anybody you share the images with will see the hdr. This only works right if you know what you are doing and if you control the image chsin completely from you to viewer. So for most people it is still best to avoid hdr if you're looking for consistency.
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