Color questions

Hello!

The concept of how colors work still confuses me a bit. I know that LUT's do not provide the exact same coloration on every monitor since LUTs are applied after ingame color information is sent to the monitor and monitors can display colors in different ways. The biggest difference is to see when you use an LCD optimized LUT on an HDT monitor. So far so good. I have only one monitor at hand so I can't really test all the differences.

Now I have made in-game screenshots of the sky color to make a comparison shot.
Vanilla game sky has a strong cyan tint.
My color rework has a more dark blue in it.

Now I made screenshots of both and copied them into Photoshop.
Looking at the pictures the colors seem to be different. The vanilla cyan sky now has a light blue color and my color rework looks more purple than blue??

Now since I can not do screenshots from what my ingame colors look like I have nothing to show you. Shouldn't have the screenshot the exact same coloration as i see ingame? Can anyone shed a light on this? ^^

Yes, every monitor is different, and every game seems to use a different colour system and applies a colour profile different from the one applied by your OS (in my case, Windows 10) to your desktop.
The game I play most is IL-2 Sturmovik 1946, which lives in the SRGB colour space. So I uninstalled the Dell high colour monitor driver and forced a Generic PnP driver with the latest IEC sRGB colour profile onto my monitor. I then calibrated my monitor with the Windows 10 desktop colour adjustment tool while looking at a greyscale ramp and simultaneously adjusting the brightness, contrast, and RGB colour controls on my monitor. So far, so good. This got me very close, but the sRGB profile which IL-2 uses is still slightly different from the one applied to my desktop by Windows. Enter ReShade. With the tonemap effect, I had to drop gamma to 0.993 and to counter the effect of Defog, I changed the RGB values to 16 for each colour. With the Adobe RGB colour space, the possible values for red, green, and blue range from 0 to 255, but with SRGB, that range shrinks to from 16 to 235. I don't know what that range is for the DCI colour space, which many newer games now use, Far Cry V for example.
At which monitor do you spend your time looking? Which colour space is it able to show? You can find this out by entering your monitor's model # into the Search Box at the Display Specifications website, www.displayspecifications.com . You will next have to find out which colour space your game uses, and you might have to use ReShade to shrink its colour space to match what your monitor is able to show.
I hope this helps, though I'm sure I've also opened up for you a very large can of worms. If you have any questions, I will happily answer them as best I can. Good luck!