After longer time I tried to calibrate my display again and create an ICC profile.
I generated profile with dispcalGUI, installed it on Windows and tried to use it with Firefox and GIMP.
Both of these programs support ICC and I could see it is working (verified with HCFR), but for some reason the colors are too off (again, but different way).
Blue is completely undersaturated which can be easily seen on pictures.
DispcalGUI report says that blue has deltaE 8.38!
I used Large testchart for LUT profiles, optimized for Gamma 2.2 with REC.709 primaries (D65).
Maybe I should use some other testchart.
Maybe my cheap display can't be calibrated more precisely... I don't know.
Please check the attached screenshots.
If anybody has an idea what might be wrong please let me know.