The short explanation
The p means progressive: every frame is a whole picture. The older 1080i is interlaced, sending alternate lines each pass, which is why it looks softer in motion. 1080p60 needs a fraction of the bandwidth 4K does, and virtually every HDMI cable ever sold can carry it.
Where you run into it
That is exactly what makes it useful. If a chain fails at 4K120 and works at 1080p, the negotiation, HDCP and the cable's basic integrity are all fine — the problem is bandwidth or capability, and you have halved the search area in one step.
Working up from a mode that works beats working down from one that does not, because each step tells you which specific requirement broke the link.
How the DR HDMI 8K handles it
The DR HDMI 8K's EDID library goes all the way down for exactly this reason: 1080p60 in 4:4:4 at 12-bit, 1080p24 in 4:4:4 at 12-bit and 8-bit, 1080i60, 720p60 and 480p60, each in 7.1, 5.1 and 2.0. Presenting a deliberately conservative EDID is the fastest way to establish a working baseline before climbing back up one capability at a time.
Want to test your own HDMI chain?
The DR HDMI 8K is sold through the HDFury.eu web shop.