Are Kyi-Leo and Toy Manchester Terrier both breeds of terrier?
yes
Kyi-Leo
The Kyi-Leo (pronounced "ki" as in "kite") is a small yet sturdy rare dog breed.
Toy Manchester Terrier
The Toy Manchester Terrier is a breed of dog, categorized as a terrier.
 The breed was bred down in size in North America from the Manchester Terrier, and is placed in the Toy Group by the American Kennel Club and the Canadian Kennel Club (the Manchester Terrier is placed in the Terrier Group.)
 Neither the Fédération Cynologique Internationale nor The Kennel Club recognize a Toy variety of the Manchester Terrier.
Scottish Terrier
The Scottish Terrier (also known as the Aberdeen Terrier), popularly called the Scottie, is a breed of dog.
 Initially one of the highland breeds of terrier that were grouped under the name of "Skye Terrier", it is one of five breeds of terrier that originated in Scotland, the other four being the modern Skye, Cairn, Dandie Dinmont, and West Highland White Terriers.
 They are an independent and rugged breed with a wiry outer coat and a soft dense undercoat.
 The First Earl of Dumbarton nicknamed the breed "the diehard".
 The modern breed is said to be able to trace its lineage back to a single female, named Splinter II. Teddy Roosevelt Terrier
The Teddy Roosevelt Terrier is a small to medium-sized American hunting terrier.
 Lower-set with shorter legs, more muscular, and heavier bone density than its cousin the American Rat Terrier.
 There is much diversity in the history of the Teddy Roosevelt Terrier breed and it shares a common early history with the American Rat Terrier, Fox Paulistinha and Tenterfield Terrier.
 It is said the Rat Terrier background stems from the terriers or other dogs that were brought over by early English and other working class immigrants.
 Since the breed was a farm, hunting and utility dog there was little to no planned breeding other than breeding dogs with agreeable traits to each other in order to produce the desired work ethic in the dog.
 It is assumed that the Feist (dog), Bull Terrier, Smooth Fox Terrier, Manchester Terrier, Whippet, Italian Greyhound, the now extinct English White Terrier, Turnspit dog and or Wry Legged Terrier all share in the Teddy Roosevelt Terrier's ancestry.
 These early Ratting Terriers were then most likely bred to the Beagle or Beagle cross bred dogs (for increased scenting ability) and other dogs.
 Maximizing the influences from these various breeds provides the modern Teddy Roosevelt Terrier with a keen sense of awareness and prey drive, an acute sense of smell and a very high intellect.
 Although they tend to be aloof with strangers they are devoted companion dogs with a strong desire to please and be near their owners side at all times. Feist (dog)
A Feist is a small hunting dog, descended from the terriers brought over to the United States by English miners and other working class immigrants.
 These terriers probably included crosses between the Smooth Fox Terrier, the Manchester Terrier and the now extinct English White Terrier.
 These dogs were used as ratters, and gambling on their prowess in killing rats was a favorite hobby of their owners.
 Some of these dogs have been crossed with Whippets or Italian Greyhounds (for speed) and Beagles (for hunting ability) - extending the family to include a larger variety of purpose than the original ratter, or Rat Terrier. Airedale Terrier
The Airedale Terrier (often shortened to "Airedale"), also called Bingley Terrier and Waterside Terrier, is a dog breed of the terrier type that originated in the valley ("dale") of the River Aire, in the West Riding of Yorkshire, England.
 It is traditionally called the "King of Terriers" because it is the largest of the terrier breeds.
 The Airedale was bred from the Old English Black and Tan Terrier (now extinct), the Bull Terrier, the Otterhound and probably some other Terrier breeds, originally to serve as an all around working farm dog.
 In Britain this breed has also been used as a war dog, guide dog and police dog.
 In the United States, this breed has been used to hunt big game, upland birds, and water fowl, and serve in many other working capacities. Yorkshire Terrier
The Yorkshire Terrier is a small dog breed of terrier type, developed during the 19th century in Yorkshire, England, to catch rats in clothing mills.
 The defining feature of the breed is its maximum size of 7 lb , although some may exceed this and grow up to 15 lb .
 It is placed in the Toy Terrier section of the Terrier Group by the Fédération Cynologique Internationale and in the Toy Group or Companion Group by other kennel clubs, including the American Kennel Club.
 A popular companion dog, the Yorkshire Terrier has also been part of the development of other breeds, such as the Australian Silky Terrier.
 It has a grey, black, and tan coat, and the breed's nickname is "Yorkie". Fox Terrier
Fox Terriers are two different breeds of the terrier dog type: the Smooth Fox Terrier and the Wire Fox Terrier.
 Both of these breeds originated in the 19th century from a handful of dogs who are descended from earlier varieties of British terriers, and are related to other modern white terrier breeds.
 In addition, a number of breeds have diverged from these two main types of fox terrier and have been recognised separately, including the Jack Russell Terrier, Miniature Fox Terrier and Rat Terrier.
 The Wire and Smooth Fox Terriers share similar characteristics, the main differences being in the coat and markings.
 They have been successful in conformation shows, more prominently in America than their homeland. Russkiy Toy
The Russian Toy (also known as the Russian Toy Terrier, and in Russia as the Russkiy Toy, Toychik,Russian: Русский той, Тойчик ) is a very small breed of dog originally bred in Russia from the English Toy Terrier which is known today as the Manchester Terrier.
 There are two types of coats in the breed: smooth coat and long coat.
 The smooth-coated variety was previously known as the Russian Toy Terrier and long-coated as the Moscow Long Haired Toy Terrier.
 Both were brought together under the same Russian Toy Terrier name in 1988 and the "Terrier" was dropped from the name when the breed was added in 2006 to the official list of breeds registered with the Fédération Cynologique Internationale and has been registered in the Foundation Stock Service of the American Kennel Club since 2008, allowed to compete in AKC companion events since 2010.
 The first official breed standard of the two varieties was written in 1966 in Russia. Terrier Group
Terrier Group is the name of a breed Group of dogs, used by kennel clubs to classify a defined collection of dog breeds.
 In general, a "Terrier Group" includes one particular type of dog, the Terrier, although other types may be included in a kennel club's "Terrier Group".
 Most major English-language kennel clubs include a "Terrier Group" although different kennel clubs may not include the same breeds in their "Terrier Group".
 The international kennel club association, the Fédération Cynologique Internationale, includes Terriers in Group 3 "Terrier", which is then further broken down into four "Sections" based on the type of terrier and breed history.