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there is no difference but there is technically no such thing as a riding 'Rink'
"Horseback-riding" and "horse-riding" are literal English equivalents of the French word équitation. The pronunciation of the feminine singular noun will be "ey-kee-tasyo" in French.
There are two basic styles of riding boots for English riding. Field and dress boots. Most dressage riders use dress boots while competing.
"Velo" in French translates to "bicycle" in English.
Stuff and stuff.
Saddle seat is a type of English riding that involves a high-stepping horse, and the rider has his/her legs farther out in front than in most English saddles. Western is a type of riding where the saddle has a horn, and is deeper and more secure than an English saddle. Western riders do not jump.
I don't believe there is a difference other than the sole. They were designed for the same thing: To protect your feet.
Il riding
In American English, that is called "bareback riding," so perhaps "horseback riding" implies the difference between riding with or without a saddle. The first horse races I went to, at a county fair back in the 50s, did not have the jockeys sitting on the horses' backs, but rather in sulkies drawn behind the horses.
You post in English riding because English riding is a "cleaner" type of riding where Western it is more of a leisurely ride.
A gerund is a verb form ending in "-ing" that functions as a noun in a sentence, while an infinitive is the base form of a verb preceded by "to" and can function as a noun, adjective, or adverb. Gerunds are used for actions or events, while infinitives are used for intentions or potential actions.
English styles of riding predate western riding. As a result, English riding is practiced all over the world, but western riding is almost exclusive to the US and N. America.