With those kind of pictures they are some times hard to keep apart, since C. bigelovii and C. fulgida can have similar looking growth forms especially at that size. Aside from the fact that when you have seen a good bunch of these plants they become easier to distinguish there is in this case another reason why this is not C. fulgida: afaik it does not occur in Joshua Tree Nat. Park, which is where the picture was taken. It does occur in the Colorado Desert of SE CA, but according to the plant list on the webpages for JTNP on the national park service website it is not documented to occur in the actual park.
Up close these plants are actually pretty different, C. fulgida tends to grow bigger (at mature size easily twice the size of C. bigelovii) more tree-like with branching often starting lower on the main trunk and generally with longer much more branching sections. C. bigelovii just does not get as tall and tends to branch from the top of the trunk often without any further branching from those initial batches. The spines while looking similar at a distance are also different with C. bigelovii having more dense spinage. Often C. fulgida will have the very characteristic chain-fruiting behavior where flowers of different years will grow from previous year's fruits, leading to it being called 'Chain Fruit Cholla', they do not always show that behavior, and it does occur on occasion in some other chollas, but not as commonly or with chains of significant length forming.
When in flower the difference is very obvious C. bigelovii has a green-white flowers (although I have seen it called yellow-green), C. fulgida has lavender to purple flowers some times with white fringes. Flower shape and appearance are very different as well and the fruits are also quite different.
The common name 'jumping cholla' comes from the ease with with sections detach from the plant, from personal experience I can say that they do appear to jump at you... (though when I got a section stuck on my arm I probably brushed it with my shirt sleeve). All chollas show this characteristic, but C. fulgida appears to be the best at it, but it is not surprising that you can find 'Jumping Cholla' used for different chollas (which is why listing plants by their latin name would make more sense). I used to have a C. fulgida and in a very strong monsoon storm I saw sections fly through the air across my patio... That and an infestation of cochineal bugs made me dispose of it.
