A quick google of 'types of kickboxing' brings up a lot of 'opinions' written on various websites. Noted that they are just opinions but it may paint a picture of general opinions from non-Axers.... And most of these versions of kickboxing don't wear silky pj's or moon boots:

Here's my links and essay:


Photo Credit kick k image by Andrey Kiselev from Fotolia.com
Kickboxing is a popular martial art that is practiced for training and competition. Like karate, there are many styles of kickboxing originating in different places and emphasizing different types of attacks and techniques. The most popular styles have affiliations and teaching studios, or dojos, all over the world.
AMERICAN KICKBOXING
A style of fighting developed in the 1970s, American kickboxing is a combination of karate and boxing that incorporates more rules of traditional boxing than many other kickboxing styles. In describing the rules, Pinellas Fitness and Martial Arts explains that punches and kicks are the strikes allowed. Opponents are only allowed to target their strikes above the waist, like traditional boxing. As with most kickboxing styles, bouts are held between fighters of similar weights and take place in timed rounds.
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MUAY THAI
Also called Thai kickboxing, Muay Thai is believed to have originated up to 2,000 years ago in Thailand. Muay Thai Online states that this more open style allows strikes not only with the fists and feet but also elbow and knee strikes. Fighters can target above and below the waist, and kicks to the legs are used to weaken an opponent's mobility and striking power.
LETHWEI
Lethwei is a raw form of kickboxing from Burma. Exceedingly brutal in its early forms, the modern competition style of Lethwei is more controlled and similar to Thai boxing in its rules. Full Contact Martial Arts describes the style as allowing attacks with all limbs, elbow and knee strikes, and even headbutts. No protective gear is worn by fighters and Lethwei is viewed by many as a more pure style of kickboxing than some of the more regulated forms.
SANDA
Sanda, or San Shou, is a style of kickboxing that stems from the ancient Chinese Wushu fighting style. The style was developed by the Chinese military, and in competition form certain strikes are forbidden and protective gear is worn. However, the Russian Federation of Traditional Wushu and Sanda describes Sanda as a complete fighting style that teaches all types of techniques including takedowns and throws.
ADITHADA
An ancient style of fighting from India, Adithada shares much with Muay Thai, according to the International Adithada Federation. Fought without protecting pads and including strikes with the feet, hands, knees, elbows and head, Adithada also includes grappling techniques. Targets on the opponents are focused on known pressure points of the body. Once repressed in India by British rule, the style is now widely practiced in rural areas of India and elsewhere.
YAW YAN
Also called Filipino Kickboxing, the name Yaw Yan means "dance of death." This style of kickboxing was developed in the early 1970s, around the same time as American kickboxing, and emphasizes fluidity of motion. Full Contact Martial Arts says Yaw Yan shares much with Muay Thai and Sanda, but differentiates itself by including many types of kicks, like the martial art tae kwon do. The style also incorporates downward striking kicks with intimidating names such as the "scorpion kick."
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REFERENCES
Pinellas Fitness and Martial Arts: Full Contact
Muay Thai Online: About Muay Thai
Full Contact Martial Arts: Lethwei and Yaw Yan
Russian Federation of Traditional Wushu and Sanda: Sanda (San Shou
International Adithada Federation: History
Article reviewed by WilliamS Last updated on: Jun 5, 2010


Links

http://www.livestrong.com/article/140262-types-kickboxing/

http://en.wikipedia.org/wiki/Kickboxing

http://forum.bodybuilding.com/showthread.php?t=687404&page=1

http://www.karateforums.com/japanese-kickboxing-v-american-kickboxing-v-european-kickbox-vt43365.html


A 2minute search brings up loads relating to the differing types of kickboxing with many references to Japanese Kickboxing, Muay Thai and K-1, so maybe fair to say that all of the sports above are versions of kickboxing so fall under the kickboxing umbrella.