B-boying is an urban name for breakdancing. People that study breakdancing from all aspects and generally live the Hiphop culture prefer to call breakdancing B-boying, simply because they fought breakdancing became to commercial and people that could do one move would call themselves b-boys or breakers. This is perticaly annoying to people that have focused all their lives on hiphop and then some guys think their bangin or cool because they can do a simple move. This is why most b-boys refer to breakdancing as B-boying.
NOTE: Street Dancers are not b-boys. Street dancing is stupid POP like dance with a street name to make it more appealing to the YOUTHS AND URBANS. Movies like you got served are absolutely fake although it does have some elements of Breakdancing, in the real world it would be b-boying not things you see Backstreet boys do.
If someone asks
Do you want to breakdance ? They are most likely not to much into hiphop itself and do not study its backgrounds lifestyle etc. DOESNT MEAN he is bad at b-boying. It could also be he just started and doesnt know the slang yet.
You wana go B-boying ? Someone that has been studying the 4 elements of hiphop and has most likely got a b-boy name for example B-boy Flash, B-boy Flex or B-boy Blaze.