Both are right. It just depends on where you are.
Just like Kleenex, Xerox and other specific brand names that
have become generic as well.
Gimme a Coke doesn't always mean someone wants a Coca Cola.
Go make me a Xerox copy...
Can I have a Kleenex, please?
So some use Cayenne as a specific pepper, like the ones in my window garden.
But I have a package of ground Cayenne pepper powder that has ingredients
of "ground red pepper". And its from McCormick or Durkee, I forget which.
I submit that for 99% of the cooks in the world, the fine line of what is Cayenne
probably doesn't matter, either.
From:
http://www.mobymud.com/spicy.htm
Interestingly, chilli is the original spelling from the Central American Nahuatl Indians and both chile and chili are derived from it.
From:
http://www.chilipepper.com/AboutChilipeppers/tabid/57/Default.aspx
Spelling and usage
The three primary spellings are chili, chile and chilli, all of which are recognized by dictionaries.
Chili is also widely used, but this spelling is discouraged by some, since it is more commonly used to refer to a popular Southwestern-American dish (also known as chili con carne, the official state dish of Texas), as well as to the mixture of cumin and other spices (chili powder) used to flavor it. Chile powder, on the other hand, refers to dried, ground chile peppers. As with the alternative pronunciation of "route" after the song "route 66", this spelling was popularized in part by the band Red Hot Chili Peppers.
Chile is the American spelling (uncommon elsewhere, [surprise surprise!]) which refers specifically to this plant and its fruit. This orthography is universal in the Spanish-speaking world, although in some parts the plant and its fruit are better known as ají. In the American southwest (particularly northern New Mexico), chile also denotes a thick, spicy, un-vinegared sauce, which is available in red and green varieties and which is often served over most New Mexican cuisine.
Chilli is the preferred spelling according to the Oxford English Dictionary, although it also lists chile and chili as variants.
From:
www.fiery-foods.com/dave/profile_cayenne.html
Cayenne as a Medicine
Cayenne is a pod type of the annuum species, and there are many cultivars, or varieties that are grown around the world. However, the cayenne you buy for use in capsules and cooking may not be made from the cayenne pod type--in fact, it probably is not. Cayenne pod types are grown around the world, mostly in Africa, India, and the United States. But in the U.S., for example, the entire crop, most of which is grown in New Mexico and West Texas, is used in the manufacture of Louisiana-style hot sauces. Virtually any small, hot red chile can be ground and placed in a capsule and called cayenne. But this is not necessarily an indictment because there is no difference in the composition of the different pod types and varieties of the annuum species, except in flavor elements and heat level. In summary, a capsule of ground piquin pods will virtually be the same in chemical composition as a capsule of ground cayenne pods. In fact, the American Spice Trade Association considers the term cayenne to be a misnomer and prefers the more generic term, red pepper.
From:
http://www.foodproductdesign.com/articles/0201sr.html
Red pepper is increasingly significant in the spice trade. Spicy capsicum pepper consumption in the United States has increased 125% over the past 25 years. In its most general use, red pepper means a ground or crushed product, which may be produced from any variety of dried, fiery capsicum pods, or "chilies."
Ground red pepper is sometimes called "cayenne." This term carries no industry standard of heat level, nor is it a particular type of capsicum. As a result, the American Spice Trade Association (ASTA) has recommended that cayenne be phased out as a product term. In the meantime, when a company feels it should continue using the term to avoid customer confusion, ASTA suggests that cayenne be parenthetical to the main term, red pepper.
See also:
http://books.google.co.uk/books?id=...&hl=en&sa=X&oi=book_result&resnum=7&ct=result
From:
http://www.atomos.net/chilli_02.htm
Chilli is made from the ground fruit of a plant in the Capsicum family. The fruits, commonly known as "chilies" or "Chillis," are fiery red or orange pods which rarely grow to more than 4 inches in length. The ground product ranges from orangered, to deep, dark red. According to the American Spice Trade Association, "Chilli" is the preferred name for all hot Chilli spices. Cayenne Chilli is another name for the same type of product. Some manufacturers use the term Cayenne Chilli to refer to a hotter version of Chilli.
From:
http://www.greenpapaya.org/category/herbs/red-pepper/
Don’t Call It Cayenne
The term cayenne comes from the Caribbean Indian word kian. Today Cayenne is the capital of French Guiana. But ironically, only a tiny fraction of the u.s. red pepper supply comes from South America or the Caribbean. Most comes from India and Africa. Tabasco (Louisiana pepper) grows along the Gulf Coast of the United States. Because so little red pepper comes from around Cayenne, the American Spice Trade Association considers cayenne a misnomer and says this herb should be called red pepper.
From:
http://www.freepatentsonline.com/5804239.html
The capsicum oleoresin ingredient of the composition of this invention is an isolate from plants of the Capsicum family, such as Capsicum annum and Capsicum frutescens. Available sources of capsicum oleoresin are commonly known as paprika, red pepper, chili pepper, and chile powder. Ground red pepper is sometimes referred to as "cayenne," to signify a ground red pepper product of extremely high heat, however, the word "cayenne" does not carry an industry standard of heat level nor is it a particular type of Capsicum. In the spice trade which is the major user of red pepper varietal and origin distinctions are being de-emphasized in favor of standardizing by heat level. Heat level can be expressed in ASTA (American Spice Trade Association) units equal to parts per million of capsaicin measured by high pressure liquid chromatography, or in organoleptically determined Scoville heat units (SU), whereby 1 ASTA unit=15 SU. Pungency levels of ground red pepper typically range from 300 to 1,000 ASTA Heat Units corresponding to 4,500 to 60,000 SU.