PEX vs PE coiled pipe for underground use
"Black polyethylene" seems to commonly refer to HDPE pipe, which is pretty comparable to PEX.  "Just" PE pipe is cheaper and less durable than HDPE - I'd advise against that underground.

PEXa ("Uponor/Wirsbo", "ProPEX", "F1960") is the best option, especially underground and in places with a potential for freezing, since it simply expands and contracts instead of bursting and leaking.  However, PEXa is a bit more expensive and requires "expansion" style fittings, which while they are most reliable and free-flowing of all PEX connections, typically require more expensive tools that may not be worth it for a one time job if you can't rent them. 

PEXc is the more common and typically cheapest form; connections are commonly "crimp" or "clamp" style which are more restrictive than expansion style, but the tool tends to be cheaper.  PEXc will also expand and contract, but to a far lesser degree before risk of splitting.

Ultimately, PEX is the best option these days, but the type of PEX is more dependent on your situation.  


If you plan to do more plumbing in the future and/or the pipe could be subject to freezing, PEXa along with Milwaukee's expansion tool (Milwaukee #2432) is the way to go for worry-free connections.  There are also hand tools available for about half the price of a used Milwaukee kit ($160-200 on eBay at the time of writing).
If you can't rent or find the expansion tool for a reasonable price, or are simply more budget minded, PEXc using crimp or clamp style connections is likely the best path.  Home Depot near me rents the tool for $16/day.