In the ’30s there was an inexpensive product called “American Cheese”; it was white or yellow and came in blocks of, I think, 1 lb. I don’t remember the sliced product being available until sometime after WWII. A somewhat more expensive,  and much superior, product was  available called “Cheddar cheese”, and more commonly known as “Rat Trap Cheese” – it always came in wedges cut from a wheel of the product. As I remember it ranged from white to yellow in color, was more or less crumbly, and ranged from mild to sharp with the whitest being sharpest. Cheddar was never known as “American cheese” – although, I assume that all or most of the cheddar we ate was of domestic origin.  Now-a-days we make cheddar, some of which is every bit as good as the best English cheddar.
 
At some point after WWII the laws changed and the word “cheese” was dropped from the “American” product – after all, the  stuff never was a real cheese by any stretch of the imagination.
 
But, cheese or not, I have fond memories of the stuff on grilled cheese sandwiches from the school cafeteria from as recently as the late 1990s. The best ones were allowed to “age” for an hour or two in the cafeteria steam table before being served… 
 
Cheers, John