A lot of this can be put down to lost in translation.
Link to Brat Style;(Click on Custom bike), Is a company, (not a specific style of bike), in Japan, that makes these bikes. Japan doesn't allow hard-tails so these guys have built bikes ranging from trail across the range of cruisers, cafe, to hard-tail lookalikes and they are from Brat Style.
Once a style craze is born then it becomes a life of its own and the original meaning is lost to the craze.
Bobber's were a modified, (To cut weight off the bike), version of bikes in the 40's and 50's Because bikes in those days were of a ridged frame people now relate them to a hard-tail.
Choppers were a cut/altered/rebuilt frame to replace the shocks and install a hard-tail and usually rake the front and these bikes started there life in the 60's and 70's.
Now any bike that has a frame cut is called a chopper/bobber and any hard-tail is called a chopper/bobber and any bobber/cafe style of bike is called a Brat.
If the styles cannot be defined as they were when they became the style they were in their time, how can so many get so offended of defensive about something that is called something, when they don't really know if they have it or not.