Officially there is only one "breed" or one "type" of Chihuahua.  Within the Chihuahua breed there are two official varieties of coat lengths according to the AKC (and other dog kennels) breed standards; the long-coat and the smooth-coat.  It is the different coat varieties that people commonly refer to as being different types of Chihuahuas, and that can be confusing because the AKC only recognizes one type of Chihuahua.

 

The Chihuahua breed has the most variations of any other breed in regard to their shapes, colors, and coats.  While Chihuahuas only have two coat lengths (either long-hair or short-hair), they come in a wide array of colors; fawn, brindle, sable, chocolate, black, tri-colored, rare blues, and rare whites just to name a few.

 

When it comes to a Chihuahuas body there can be big differences from one dog to another.  Some Chihuahuas have longer bodies, with long slender legs, while others have short compact (cobby) bodies with short legs.  These differences can result in some Chihuahuas being over a foot tall, and others as short as 4 inches.

Unofficially the term "type of Chihuahua" is used way more loosely and is used to refer to distinctions in the breed like coat length, head shape, and body type. Using the unofficial standards, there are many different types of Chihuahuas.

 

Why don't they just separate the Chihuahua into two breeds - The Long Coat and the Smooth Coat?

There are many reasons why they don't and why they can't separate them into separate breeds.  The first and foremost being that they are technically still the same breed.  Another reason is that when you breed two smooth coat Chihuahuas together it is entirely possible to get a long coat offspring.  This lack of breed predictability disqualifies the Chihuahua from being considered for separate breeds.