My first thought would be to see what the differences are between your configs for LR3 and LR6 regarding default camera profiles (Camera Calibration panel) and any default presets that the import process applies.
Well, that sounds logical and that's what I tried first. What makes me worried is that the 2012 process and Adobe Standard setting of calibration needs 25-50% Vibration/Saturation boost to get colors of gray, rainy days - same NEF file, same screen panel, same default setting of old and younger LR. That looks to me too wrong to be a standard feature/failure of software, I still suspect I do something wrong, just can't find what.
Thanks for for any idea or hint - what default presets or other options may be behind?
Try turning off the GPU in LR / Edit / Prefs / Performance. Using the GPU is new to LR 6 and maybe the video drivers aren't configured properly. Turning it off should rendering things more like previous versions of LR at least with the same settings. The GPU is only used in Develop, so if the colors are washed out the same in Library and Develop it's likely not the GPU being on that's affecting things.
If that's not it, then can you share your raw file, and post screenshots of what the LR 3 default settings look like and the LR 6 default settings look like. Upload the raw file to www.dropbox.com and post a public share link, here.
You can use older processes. 2010 and IIRC 2003 and see if that is more to your liking. But you will lose some of the LR added features by using the older processes.