I myself didn't inspect the howstuffworks link, so I'll critique based on my knowledge the the various weapons books I have:

Mangonels (medieval) and Onagers (roman) usually had either a scoop or a sling as their head. The scoop was far easier to build, but the sling was better and more common (adding as much as a third to the distance). I can't find pictoral references to the head being a bowl, but they may be out there, and it doesn't really matter anyway, unless you're doing a sim.
Also, they came both wheeled and staked to the ground. If the artillery was defensive, it'd probably be staked so there was less recoil (and less strain on the pieces, less repositioning, etc).
They were powered by a wound skein, probably made of sinew. It would be threaded around the catapult arm, and wound by two winches at the base of the arm, until it was ridiculously tight. An additional winch would be at the rear of the catapult, and would be wound to bring the arm down to loading position (and put the arm under even more tension).
The projectile would be loaded, and the trigger pulled, and the arm would come flying up, hit the cross-bar, send the rear wheels of the mangonel or onager into the air (hence the nickname of wild ass), and the projectile would be launched.

Here's a good site:
http://www.redstoneprojects.com/trebuchetstore/build_a_catapult.html

Thanks for the info! That site could come in handy.

All artillery played primarily a role as seige weapons. It was not common (or possible) for larger engines to be deployed in battle, though smaller versions certainly were.

Artillery also was far more important to an attack than defender. Seige weapons fired from inside a city could do relatively little damage to a force that has surrounded a city and fortified itself. On the other hand, seige weapons fired INTO a city wreak havoc, because anything they hit becomes a problem. So they were primarily offensive seige weapons.
Ballistae were just better forms of the Onager or catapult, they were designed as offensive or defensive. Ballistae could throw a projectile twice as far, and far more accurately, than a similarly sized onager.

I also agree you don't need such great historical accuracy, but there needs to be an effective way of the catapult working. Just some minor modification of adding a couple winches and rope-covered-bars is probably enough. The animation is fine.