Why "horseback riding" and not simply "horse riding"?

Solution 1:

It's probably just a conflation of the phrases "horse riding" and "on horseback". Couple that with the fact that you are, generally, riding on the horse's back, and you've got a recipe for common usage.

Solution 2:

I suspect it's left over from the days when a ride involving a horse could be done on the back of a horse or in a wagon or carriage behind a horse.

Solution 3:

It's a good question. As TomH wrote, horseback riding is a primarily American usage. It's not in English generally.

I have not heard anyone in the UK use it. The only time that I heard it was in American films. I remember it because to me as a British person, it sounds odd and unnecessarily long. I remember thinking to myself, “Why say horseback riding? What other part of a horse do people ride on?”

In the UK, it is obvious that riding a horse means riding on its back.

Use of the word back in this is just an unnecessary extra. Whenever it is talked about here in the UK, people just say horse riding. It is also enough to say riding, because it is obvious what is meant by the context.

The difference between the two ways of saying it depends on the regional flavor of English being used. Horseback riding is used by people speaking American English, and horse riding is used by people who are not speaking American-style English, including English and other British people.

This explains it: http://dictionary.cambridge.org/dictionary/british/horse-riding?q=horse+riding