"Marketed". That's the key word. When a manufacturer of an intake dynos a car, it has a huge fan blowing in the front of the car and usually with the hood up. The engine bay doesn't get that hot during the runs. Also, an intake that is claimed to have made power for one application may not have been dynoed for another application but they still "market" that type of intake as making gains.
I've never heard of an AEM "short ram". The AEM intakes for this car are all "CAI's" with the filter behind the fake airduct.
You seemed to have missed that whole business about hot air being drawn into the intake. Most short ram intakes will show gains UNTIL the engine bay fills up with HOT air, then you will LOSE power. It's just physics. Short ram intakes are not necessarily Cold Air Intakes. Intakes like the K&N FIPK and Comptech are short rams but they have a box that surrounds the filter which keeps out the hot air.
This subject of "lid on", "lid off" has been going on since the first S2000 got into consumer hands. With lid on, the intake still sucks in hot air. With lid off, it will suck in hot air sooner and more of it. Do what you like. It's personal preference.

 

http://www.aempower.com/product_intake.asp

Scroll down. You'll see the Short Ram. It's what I had on my Prelude.

From what I've seen of the AEM V2 applications on Prelude's, it is basically somewhat of a replica of the Short Ram in that it sits inside of the engine and doesn't bend or dip down below the engine area to bring in colder air.

 

Interestingly enough, I found this on their site too:

So does that rule out getting a CAI if you're Supercharged?

 

As well as:

Which is what I thought it would do being that I will pull in air from the engine compartment just as a short ram would or even a open box stock intake... But, I'm not an expert by any means. I'm all new to this...

Edit: It seems that I misread that...

 

You don't want to put one of these short rams on an S2000. There is no point to it. Why pay that kind of money when leaving the OEM lid off does the same thing?

Not necessarily. A CAI for a supercharger is always a good thing but is of minimal advantage. For a non-intercooled S/C, the pressurized intake charge gets very hot anyway, so a slight difference in the original intake temp won't be a great difference by the time it gets to the manifold. For an intercooled S/C, the intercooler is efficient enough that, once again, any difference in original intake temp won't manifest itself in a significant difference at the manifold.
BTW, both the existing supercharging kits for the S2000 has it's airbox bringing "cool" air in from the hole at the right side fender wall. This air comes from the area behind the fake airduct. However, a few owner's with S/Cs have used filters clamped directly off the blower with little loss of power output. Personally, I prefer to draw my air for the blower from that hole in the fenderwall.