The short explanation
Digital Cinema Initiatives P3 sits between Rec.709 and BT.2020. It is noticeably wider than the old standard, particularly in reds and greens, and it is achievable — good modern displays cover most or all of it.
Where you run into it
Whether you should tell a source your display wants DCI-P3 depends entirely on the display. Get it right and colors are accurate. Get it wrong and the picture is over-saturated with a red tint that is most obvious on faces.
The interesting case is a source with no DCI-P3 output option of its own. Telling it the display wants DCI-P3 through the Dolby Vision data block makes the LLDV tone mapping output DCI-P3 colors anyway — a genuinely useful trick, and a genuinely bad idea on a display that cannot handle P3.
How the DR HDMI 8K handles it
DCI-P3 is a selectable primaries option in the DR HDMI 8K's Dolby Vision section, and one of the built-in custom EDID tables is an LLDV DCI-P3 profile with a BT.2020 counterpart. Switching between the two and looking at faces is the quickest way to find out which one your display actually wants.
Want to test your own HDMI chain?
The DR HDMI 8K is sold through the HDFury.eu web shop.