Correct. Kobe beef is from a specific region. Omi beef, Miyazaki beef, Kagoshima beef, Hokkaido beef, Matsuzaka beef, Sendai beef, Saga beef, etc. are all different. Though the nuances between some aren’t always super apparent, sometimes you can tell. E.g. Hokkaido beef is less marbled, even for the same A5+ rating, and I think Kumamoto beef is even leaner tasting. Omi beef and Matsuzaka beef are my favorite. Omi being the best balance between beefiness and taste of fat for my preferences, and Matsuzaka is best for grilling if you want a real luxurious taste. I haven’t had Kobe grilled, only shabu shabu. The best depends on how you’ll cook it, but by and large (and again, it depends on the cut, not merely the provenance of beef) - the highest regarded are Kobe, Omi, and Matsuzaka.
Now, according to an article from Honolulu dated September 2015, only 3 restaurants in the US were authorized to serve genuine Kobe beef: Teppanyaki Ginza Sumikawa (aka Teppanyaki Ginza Onodera) in Honolulu, SW at the Wynn in Las Vegas, and 212 Steakhouse in New York.
As of July 2016, apparently there’s only 9 official places in the US serving real Kobe beef (8 restaurants - Alexander’s counts twice because they have two locations serving it):
However, there are quite a few, perhaps several dozen restaurants that serve real wagyu that’s not Kobe, but usually from Miyazaki or Kagoshima. In California alone, I’ve probably had it at over a dozen places, to varying success. Californios did a great “swagyu” barbacoa with pineapple juice, and recently a delicious Kumamoto wagyu carne asada with leek oil. Aubergine in Carmel has a signature wagyu dish that’s washed in sake, soy, etc. and wrapped with nori (usually we’ve had it from Miyazaki). Yakiniku Yazawa had an incredible Miyazaki “Yazawa Yaki” that was one of my favorite bites this year. I’ve also had several Miyazaki or Kagoshima wagyu dishes that weren’t much to write home about, but probably due more to how they were prepared.
One place did serve me the scraps, as a side so as not to waste anything, though. Flambeed in Macallan 18, but more for the theatricality of it because it kind of overcooked the beef. I prefer the scraps in a fried rice with egg. Never liquified for a hot dog, and I really think nobody in his or her right mind would ever do that.
There’s also “washugyu” - wagyu-style beef that’s not actually from Japan, but from like Idaho or Australia. And there’s also “F1” (very popular in Japan), which is mix of wagyu and other breeds like angus or holstein.