The problem--or at least a problem--is that what constitutes karate is not necessarily so clear. In the first place, the Okinawan system, or more accurately systems, of martial arts that came to be fairly widely practiced in Japan proper as 空手 by the beginning of the twentieth century is and always has been itself a mixed art. On Okinawa elements of Chinese martial arts like the White Crane Style were combined with native practices. When Funakoshi Gichin brought Okinawan martial arts to Japan he called it 空手 and established, along with many other changes, such things as the 二十訓, Shoutoukan's "Twenty Precepts," which integrated Japanese philosophical and martial arts traditions with the Okinawan style, giving such well-known principles as 一、空手道は礼に始まり礼に終る事を忘るな, "Do not forget that karate begins with a bow and ends with a bow." Things like colored belts to show rank are a development of 空手 after it came to Japan, borrowed from Juudou. Lineage is often stressed by karate schools today, both in America and without, but many of the most common karate styles do not derive from Funakoshi's 空手, which eventually became Shoutoukan. "Karate" was "brought" to Japan several times from Okinawa, and then again to America and elsewhere. Indeed, many styles, such as my own Shitou-ryuu (which takes its name from Mabuni-sensei's own masters' in his original styles), were already in Japan in the early twentieth century describing themselves as mixed styles.

At the same time, "karate" was never homogeneous. Modern karate schools claim descent ultimately from one (or more) of three Okinawan sub-styles: Shuri-te, Naha-te, and Tomari-te, named after the locations that are claimed to be their birthplaces. Individual masters carried their understandings of these particular sub-styles to Japan, where they developed into many different styles--the lineages of these schools was, from an early date, disputed, and often tradition alone is the only way we can find the "real" descent of any particular style. Styles developed into new styles, teaching techniques changed, and schools combined with each other as masters learned from one another or learned additional styles. As a result, "karate" as practiced by different styles was from very early on very different, even as most schools laid claim to the same basic ancestry and to the same (or similar) precepts as were set out by the earliest masters to bring the art to Japan. So, for example, Goujuu-ryuu has only 12 kata, while Shoutoukan today has 26 or 27 kata (depending on the school), Shitou-ryuu has around 60 (the exact number depends on the branch), and Okinawan master Motobu Chouki is often, probably erroneously, said in doujou today to have known only a single kata. Likewise, you'll find the curriculum completely different across schools or even doujou, and this is even without getting into actual techniques.

Many styles of karate have been developed in America, have been developed for American pupils, or have been developed with American influence, whatever that means exactly. Whether any of these constitute "American" karate is harder to say. It is somewhat disputed who first brought karate to America (probably not least because it's not always clear what 空手 is exactly), but Robert Trias established a school teaching something he called "karate" in 1946 and founded the now-defunct United States Karate Association as early as 1948. Trias is a typical example, and he'll do nicely. Trias claimed (the uncertainty on my part here does not necessarily mean anything--I've never bothered to check his lineage, and already in the late nineteenth century masters were claiming lineages that could not be verified) to have trained with Chinese and Okinawan masters while in the Navy, apparently learning some Chinese martial arts, as well as Shuri-te and Kenpou. Trias' style is now known as Shuri-ryu, but was originally known as Shuri Karate Kenpo, Shorei-Goju ryu, and Goju-Shorei-Ryu. The odd changes in name, however, should not raise eyebrows. Trias could not decide on a name himself, but neither could Shitou-ryuu's Mabuni-sensei, who originally called his style Hanko-ryuu. Like many Japanese styles, Trias' style was derived from many different sources, which were not all Japanese or Okinawan karate, and not necessarily karate at all. Which raises a couple questions. In the first place, while many other karate styles are also mixed styles, typically their founders attained mastery with masters in several different styles, before combining their mastery into a single style. Trias, and other American martial artists like him, could not always claim to have studied formally with masters the way someone like Mabuni-sensei could. Whether that means his style is not "really" karate is another thing--masters like Shimabukuro, whose Isshin-ryuu was taught to the Third Marine Division on Okinawa and brought back to the United States in modified form, had similarly uncertain lineages, and certainly borrowed from styles that many organizations maintain they had not necessarily formally attained mastery in. Additionally, in many ways Americans like Trias could be said to be returning to Okinawan karate. Actually claiming that would be absurd, but the interest that American masters had from early on in martial arts outside of those descended from the three Okinawan sub-styles (themselves mixed styles) is notable. In some cases that interest turned these masters towards Chinese or other Asian martial arts that had influenced or were otherwise related to karate, while in other cases it turned them towards Japanese and Chinese martial arts that had developed in the meantime, sometimes themselves influenced by karate. So we have things like American Kenpo, which Ed Parker originally called Kenpo Karate but which, in its current form, does not actually currently use the karate label.

But karate was not brought to the United States all at once or in a single place. Prior to Trias karate was being taught in Hawaii, most notably by James Mitose, who taught a somewhat amorphous martial art that had many names over the years. Mitose is very controversial: he taught something between Kenpou and karate, taught only a single kata (Naihanshi), and in a somewhat bizarre court case was convicted of murder and sentenced to life in prison. But already by Mitose's tenure as a teacher of martial arts the very mixed nature of karate and karate-influenced styles developing in America was apparent. Styles that are thought of as being "American," that claim development in America from American (or Asian immigrants) masters, or that call themselves "American," whether they actually call themselves "karate" or not, are typically thought of as being mixed--or, to use the more proper terminology, "eclectic"--styles. But I hope it should be clear by now that there's really nothing all that strange about that, except perhaps the very concentrated contact that certain American masters had with particular forms of Chinese, Okinawan, and Japanese martial arts, often over a comparatively short period of time. Likewise, it's not even clear who says what constitutes "karate." Different karate associations have different definitions and lists. The World Karate Federation, the only karate federation recognized by the IOC, recognizes last time I checked 8 karate federations worldwide, including the Panamerican Karate Federation, which these days is strongly associated with UFC. The Japan Karate Federation, meanwhile, only recognizes four styles: Wadou-ryuu, Shoutoukan, Shitou-ryuu, and Goujuu-ryuu. This is because the JKF was founded from a unification of the Japanese karate organizations that existed in 1964. Even more specific organizations can't decide on what constitutes "karate." The Japan Karate Association, now a member of the JKF, is a Shoutoukan karate organization formed by some of Funakoshi's original students. Yet it's had something like six or seven splinter groups who can't agree on who's actually practicing karate and who's actually in charge of various styles--importantly there was from pretty much day one a dispute as to whether competition should be allowed in Japanese 空手.