The short explanation
Ultra High Definition is 3840 x 2160 — exactly four Full HD frames. Cinema 4K (DCI) is 4096 x 2160, slightly wider. In everyday use 4K and UHD are treated as the same thing, and for consumer content they effectively are.
FUHD, Full Ultra High Definition, is the 8K step: 7680 x 4320.
Where you run into it
Mostly it matters when reading specification sheets, where a device may list UHD and DCI separately. A display that accepts UHD does not automatically accept the wider DCI variant, which occasionally surprises people feeding it from a computer or a cinema player.
How the DR HDMI 8K handles it
The DR HDMI 8K lists support for SD, HD, Full HD, UHD and FUHD, including DCI formats and DSC. Its EDID library covers the UHD range in detail, from 4K120 down to 4K24, so you can advertise a specific UHD capability rather than a vague one.
Want to test your own HDMI chain?
The DR HDMI 8K is sold through the HDFury.eu web shop.