I've never really thought about whether the game looks the same using its Windows vs Linux renderer. I installed HL2 to check out the anniversary update and decided to try the Windows renderer on Proton. Would you look at this difference... Linux is the top one, Windows is the bottom:
Is this is a thing? Have you tried other games and seen the Windows version looking better?
Old games don't use "HDR" the same way we use it today. In old games, enabling HDR makes the lighting calculations in the game engine have infinite range which will then be mapped onto SDR colorspace, which is all software and very much supported in Linux.
If anything the screenshots show a gamma calibration issue. From my experience on Linux native Team Fortress 2, the in-game gamma slider does not do anything.
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A great, short explanation of the different meanings of HDR is up to minute 1:40 in the video below.
https://youtu.be/frBNnNWEbyk