Should be interesting to see this area after the fire last fall. Remember being in Yellowstone in 1989 after the fires the Summer after seeing Nature at work. Note: Should be Dilly and not Dilley Cliff Trail.
I'm sorry they had to close it but not surprised. When I hiked it in October 2014 one sign said it was "Closed" even then, but there was no barrier of any kind and a second sign said "Experts Only," so the first sign wasn't really serious. The majority of the Trail was an all-fours scramble, like this:
As for the name, both of those alternatives appeared, on different signs: