Wagyu vs American Wagyu
According to the American Wagyu Association, most Wagyu beef are bred with Angus cattle in the United States. Thus Wagyu in America are half-blood. There are full-blood cattle, but most of the full-bloods are used for breeding, not for eating purposes since the numbers of full-bloods are very limited. The Association was initiated in Texas on March 14, 1990. Since then, it promotes Wagyu into American and Canadian markets and controls the registration of Wagyu mainly in those countries.
Even if it is an American Wagyu or a half-blood, as long as they are one of those four breeds, they would be called and registered as Wagyu. However, you may notice that some specific prefectures’ (like Kagoshima and Miyazaki) names, grading for the meat such as A5, or a specific name of the brand are written on the menus in restaurants, implying that the Wagyu is imported from Japan. Moreover, those imported ones are considered as the more premium ones -- they have passed the strict tests. For example, “American Kobe” does not exist. That is the same thing as saying “Japanese California”, which makes no sense. Kobe is used as a marketing term; Kobe “style” implies that they are not authentic Wagyu. It is the same how American Champagne cannot exist. All sparkling wine cannot be called Champagne, which are produced in a specific area in France and meet the highest standards -- just like Wagyu.