The Cooper GFCI in the video appeared to be new when the video was made (in 2014) and the light was only on when the GFCI was tripped.
The newer Levitions have a green LED that stays on as long as it's working, and turns red when tripped. I believe the Pass and Seymour made units work like the Coopers, where the light is only on when tripped.
Some older GFCIs in the 70s and 80s seem to have the a trip indicator also, but not all of them.