The short explanation
Digital Cinema Initiatives defines the formats used in commercial projection. DCI 4K is 4096 x 2160; consumer UHD is 3840 x 2160. Same height, 256 pixels difference in width, slightly different aspect ratio.
The name also attaches to DCI-P3, the cinema color space, which is a separate thing entirely.
Where you run into it
Mainly with projectors and computers. A projector may accept a DCI timing where a television will not, and a graphics card can be persuaded to output one. If a source offers a 4096-wide mode and the display refuses it, that is the distinction biting.
For ordinary playback it rarely comes up: streaming and disc content is UHD.
How the DR HDMI 8K handles it
DCI formats are among the supported signals listed for the DR HDMI 8K, alongside DSC. If a DCI mode is not being offered, its EDID readouts show what the display is actually advertising, which settles whether the mode is unsupported or simply never announced.
Want to test your own HDMI chain?
The DR HDMI 8K is sold through the HDFury.eu web shop.