I understand your situation and confusion so I found out the empirical answer for you as I have both components at my house.
Using a PoE capable Cat6 network cable, I connected the US-8 on port 1 to one of the 4 PoE capable ports on the US-8-60W and the US-8 will not start as expected. I was only able to get the US-8 up and running with the supplied power supply. I hope this takes away any doubts you may have had.
The US-8 (802.3at) expects a different PoE standard than the US-8-60W can supply (802.3af). The Unifi nanoHD, on the other hand, should also be able to be powered with 802.3af according to the manufacturer's product page.