As far as I know, cougars are referred to as panthers, mountain lions, and just cougars (among many other things) all over the US. I think that the term "panther" is most correctly used to describe cougars, but I could be wrong. I rarely hear leopards and jaguars called panthers, not counting "black panther" variations.
For the main point of this thread, I think that it shouldn't be necessary for jaguars. I have seen a jaguar climb an enrichment platform at a cat rescue near me, but ultimately, at least compared to leopards, they aren't huge climbers. I mean, if you put a human in an enclosed area without that much to do they'll probably climb a tree/climbing frame in their spare time. Even in the "wild" humans still climb things for fun. But compared to our chimp and bonobo relatives, we don't climb much, nor is it essential.