Yes, Cayenne is a somewhat generic name for any hot pepper that contains Capsicum. Capsicum is part of the botanical name used for formal classification of plants, in this instance, pepper plants. So in a way, Capsicum is part of the official name for a broad family of peppers, Cayenne is sort of a nickname that speaks to the same broad family and this nicknaming convention is a characteristic that speaks to a time long ago when herbs in general were more widely used by people, and differenet locales of people tended to develop their own nickname for the same herb. Lobelia is another good example; many people didn't call it Lobelia, they called it Puke Weed because this is one of the beneficial effects that this herb produces when it is taken in a large dose. It seems that one of the reasons for there being many nicknames for the same basic herb is that the old fashioned use of herbs by people in some instances (proably not all) predated the latins and or nerds in white coats who were commisioned to go into a lab, tear an herb to pieces, separate into it's constituent isolated compounds, and then establish an official-sounding name for the plant under the guise of "regulation". This official name often speaks to one of the various compounds discovered, what white coats like to refer to "the active ingredient"......ergo "lobeline" is a main compound found in Lobelia, and "Saffrole" a main compound of Sassafras.
The thing that makes this a bit confusing is that there is a specific variety of pepper known as Cayenne, with the official botanical name of Capsicum Frutescens....which is also the same name used to speak generically about any hot pepper that contains the common natural compounds/ingredients found within the plant/fruit. Different people also have different ways of pronouncing even this official name. Some call it Capsishum, others call it Capsashum and other people probably have other deriviate names they use.
Cayenne, Habanero, Jalapeno, Chili, Cherry, Spanish, Mexican and other hot peppers all belong to the Capsicum family. Just like with other herbs, various locales around the world have their own local nicknames names for peppers that belong to the Capscicum family. For instance, Habanero is known in some places as "bird", "African bird" as well as "Scotch Bonnet" pepper. Habanero is generally considered the hottest of hot peppers, and the African Bird variety is said to be the hottest of the hottest of hot peppers , and in a pinch, someone looking for "Cayenne" for medicinal use, and is not able to find it, can substitute Habanero and or Bird Pepper and or .... you get the idea :)
PS - as to the original question, perhaps they are scarce around L.A., but Cayenne Peppers do exist. Since Schulze was once located in the LA area (I dunno where he is now, he may still be out there) and since he was a target of the FDA/AMA syndicate, maybe this in part owes to the apparent scarcity in yoru area. Anyway, rest assured, Cayenne still does exist in it's many variations. In fact, no less than 3 months ago a fine crop of them existed in my garden (several varieties) just before I harvested them. Some of them have been dried and turned into "pepper" - the kind that can be shaken over top food or put into tea. Some of them have been tinctured. Some of them were put in the freezer so as to have some whole/fresh on hand for any need that may pop up that calls for whole/fresh.