I don't think the record company automatically holds the rights to live shows. Bands usually sell or rather license their product to the record company (who pay for studio time) who then can do with it whatever they please. I'd assume BJ's tours have a fair share of "powered by" sponsors, but I doubt they own the live shows themselves

Of course, all of this is unless the record company specifically chipped in for a show that was set to be recorded - of course the record company owns One Wild Night Live. Since these videos were professionally recorded, who knows? I believe some previous live videos have the "Copyright the Island Def Jam company" bit at the end. With broadcasts, these parties might also hold rights.