You should never use the display profile as a document profile!
The document profile should always be a standard color space like sRGB, Adobe RGB or ProPhoto.
Using the display profile at document level disables and cancels out all color management. All references disappear, and the file is displayed according to whatever characteristics your display has. That is a very dangerous thing these days when display characteristics vary enormously. As you have experienced.
With correct color management, the data are converted from the document profile and into the monitor profile, and these corrected values are sent to screen. You need both these profiles, in their correct places. The display profile is set up at system level and used by Photoshop on the fly, as you work, without any user intervention.
To get absolutely correct colors, you need to use a calibrator to measure your display and write a profile accordingly. A generic system profile can never be any better than a rough approximation.