saddle
That would be called a saddle
cavalry
Formally, an "equestrienne". Generally, all horseback riders are called "equestrians".
Jousting !! ~
A rosette is what we call the ribbons you win while horseback riding.
You would call him a Jackaroo.
They call it horseback riding & yes we consider it as a sport.
Hunt seat. western. saddleseat. sidesaddle. jumping. trail. dressage. gaming. roping. reining.
Jackaroo
It's called a crop, or a riding crop.
For Howrse: Jackaroo
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.