First Peoples Buffalo Jump State Park is around 200 miles southeast of its Canadian counterpart and was known by "The former name of the park was derived from the Blackfeet word "Pis'kun," meaning "deep kettle of blood," and the nearby town of Ulm".
Perhaps this explains the name confusion over the years, and why the Canadian site might be hesitant to use a Native name of a site relatively nearby in Montana that used to go by that name (or similar).
Interestingly enough, the Montana site just became a National Historic Landmark this week, and now officially has national protection. Perhaps a worthy extension? Indeed, the size, age, and fossils seems to be fairly comparable between the 2 sites. On a side note, I see Lewis & Clark were mentioned by Solivagant, interestingly the site(s) he described are sites found in Montana, not Canada.
The criteria for inscription for the Canadian site starts with, "Head-Smashed-In Buffalo Jump is one of the oldest, most extensive, and best preserved sites..."
Now I know why they said "one of the".