Oh this one is going to rattle some prams...
I think in the USA bellydance met burlesque thanks to Sol Bloom and Little Egypt and all that World's Fair stuff, and has been trying to disentangle itself for some time.
I'm no expert but my impression is that burlesque uses all sorts of musical/dance references but the point of the act is the teasy stuff. I don't think a traditional burlesque audience would have given two hoots about the performer's dance technique as long as it made her look good. "Bellydancer" is an good one - a fantasy of exotic mystical eastern promise conjured up with a pair of harem pants, some snakey movements and a bit of wobbling.*
But there might have been some crossover thanks to Hollywood... the Hollywood bellydancer image would have been influenced by the burlesque take on bellydance, however "incorrect", that was current in the US at the time. Then this Hollywood image got taken back to Egypt and influenced their cinema and dancers, hence, I believe, the cabaret costumes.
Whether this has had any effect on the dance itself rather than the costumes/image, I don't know. I suspect not much... Hollywood movies are not exactly showcases of dazzling bellydance technique.
* ooh, I forgot the navel jewel!!!!! (post Hays code)