I guess this was a silly question?
I agree with Norma that it is not a silly question. It is just a bit difficult to answer. But I will give it my best shot.
When Steve and the original Moderators came up with the indexing system for the forum for the different styles of pizza, the California style was a recognized one and, hence, was included in the indexing system. That style reflected the work of people like Woflgang Puck who put exotic things like salmon and caviar and creme fraiche and duck on pizzas, and Alice Waters who specialized in using locally produced ingredients (preferably organic) on her pizzas, and Ed Ladou who, as the original pizza chef at Puck's Spago restaurant and the father of the modern California pizza style, carried forward that style after he left Spago's to open up his own restaurant, Caioti Pizza Cafe, until he died in late 2007 of cancer at the age of 52. I suspect that with his death, some of the mystique and interest in the California style died with him. For examples of the types of pizzas and pizza topping combinations that Ed Ladou created, see the menu at Caioti Pizza Cafe at http://www.caiotipizzacafe.com/menu.html;
From the chain side, the California style was popularized by California Pizza Kitchen. Examples of the types of pizzas that are offered at CPK can be seen at http://www.cpk.com/menu/#original-crust-pizzas
As you may already have noted, the California style of pizza does not get much activity on the forum. Maybe it is because the use of fancy or exotic toppings is no longer limited to the California style. Many of our members, such as Craig, John Dellavecchia, John Conklyn, and Marlon (bakershack), just to name a few, have routinely used higher end toppings and cheeses and tomatoes on their pizzas. The crusts are different, but the thickness of the crusts is in line with the crusts used to make the California style. They are both on the thin side. And the pizzas are of modest size. Also typical of the Neapolitan style is the use of Italian 00 flours for the dough but no oil or sugar. The California style typically uses something like an all-purpose flour or a bread flour and oil and sugar.
By contrast, the style of pizza that is known as the American style, at least on this forum, is characterized by a thin- to medium-thickness crust. The well known chains like Papa John's, Pizza Hut, Domino's, Mellow Mushroom, Hungry Howie's, and Little Caesars, exemplify the thicker crust versions of that particular style. Some of them, like Mellow Mushroom, may get a bit exotic on the toppings but most use the typical toppings choices such as pepperoni, sausage, mushrooms, vegetables, etc., with occasional specials to reflect changing consumer preferences. The sizes of the American style pizza will range from small to extra large, and will usually max out at 16". The thicker crust versions of the American style will usually contain a lot of oil and sugar, which combine to produce a soft and tender crust.
As for the thinner versions of the American style, they are typically represented by the pizzas made by companies such as Round Table, Shakey's, Monical's, and Donatos. Their crusts will often have a combination of chewiness and crispiness. Standard flours are used to make these pizzas and the toppings choices are similar to the other American stye pizzas.
Whatever the version of the American style pizza, few will confuse them with either the California style or the Neapolitan style.
If I have forgotten anything or not fully answered your question, please feel free to ask for further explanation.