Gosh, that seems very messy and confusing doesn't it.

They only accept people with sports injuries? What sort of agency are they?

Others here can probably help you better as I'm not in your country, but I wondered...

Do they call it Acquired Brain Injury and do they offer support for people with ABI?

The BIAA have definitions.
http://www.biausa.org/about-brain-injury.htm

http://bianj.org/about-brain-injury
Brain Injury Alliance of NJ

Mayo Clinic Definition - TBI

Someone else here was talking yesterday on another thread about not getting support because their condition wasn't classified as TBI.