PS I have (with the monitor) a cabel DP <-> miniDP. Maybe it is better to choose the second?
PSS I do not understand why such a difference in price (
This is the DP version I use to get 4K@60Hz on my DP monitor: https://www.amazon.com/CAC-1507-DisplayPort-Adapter-Chromebook-Thunderbolt/dp/B018AX3HJO.
This is the HDMI version I use to get 4K@60Hz on my 40" TV: https://www.amazon.com/CAC-1504-Chromebook-Thunderbolt-Supports-3840x2160/dp/B016QO8186.
These are Amazon shortcuts; I am not a fan of the broken/used crap you get through eBay and I don't trust and most definitely won't use ALI at all.
The following cables/adapters are validated by Intel for your NUC: http://compatibleproducts.intel.com/ProductDetails?activeModule=Intel%C2%AE%20NUC#