The best way is to use a dedicated calibrator, like an i1 Display, ColorMunki, Spyder and so on. This will calibrate the display to a "normalized" response - but more importantly, it will also make a profile that describes this response in detail. That's what color managed applications use.
I really don't know what the Windows calibration utility does, I have never used it. I don't know if it only calibrates, or if it also makes a profile. This is an important distinction.
Generally, though, I usually say that if you don't have a calibrator it's better to use a standard generic profile. It won't be accurate, but it will be consistent and reliable. For a standard gamut display (99% of them) that's sRGB, for a wide gamut display Adobe RGB.
The problem you showed here is very frequently caused by faulty profiles from the monitor/laptop manufacturers, who just can't seem to get this right. These profiles are either installed from the CD that shipped with the unit, or they come through Windows Update. Manufacturer profiles should be avoided. There's usually no advantage over sRGB even if they do work as intended.