My take is that calling what he does American Jiu Jitsu is as legitimate as any other style name in martial arts. Very rarely can a "style" lay claim to inventing something new. What is different between "styles" is where they choose to put an emphasis or use their training time. Such as TKD emphasizing kicks vs. many other styles of karate or Judo emphasizing throws vs. BJJ emphasizing ground work.
A good Judo guy is going to know plenty of ne-waza but his throws are going to be better because that is his emphasis - a good BJJ player is going to know throws but his ground game is going to be better because that is where he puts his time.
AJJ is saying that they spend more of their emphasis/training time on wrestling/takedowns than a typical BJJ school (and I don't think they really do any gi work at all). It is all about where you spend the limited amount of training time you have. A style/name is just a convenient label that let's people categorize things and let's the label's creator/owner differentiate himself and market more effectively.
Visually it might look like this (don't critique the artwork I took like 5 minutes ;-)):
Peace,
Zen Mojo
zenmojobjj.blogspot.com/
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Get your grips, use your shoulder for pressure and whisper something dirty to the mat as you pass...