The dictionary definition of a sport from dictionary.com:
Sport[ spawrt, spohrt ]
1. An athletic activity requiring skill or physical prowess and often of a competitive nature, as racing, baseball, tennis, golf, bowling, wrestling, boxing, hunting, fishing, etc.
2. A particular form of this, especially in the out of doors.
3. Diversion; recreation; pleasant pastime.
So by the definition of the word, it's a sport. Physical in nature, competitive, outdoors, etc.
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That definition won't hold up in any of the Sport Federations. At least not item 3 and item 2 won't get too far either.
Let's try something closer to what the real world sees a sport has from a internationally recognized English Dictionary.
Oxford says: (as a noun since that's how it's being used here)
An activity involving physical exertion and skill in which an individual or team competes against another or others for entertainment
But I like Webster's since it gives the key requirements that most organizations will look at to accept something as a sport.
"a contest or game in which people do certain physical activities according to a specific set of rules
and compete against each other"
I doubt you'll get any sporting venue or organization to buy off on "3. Diversion; recreation; pleasant pastime." as what's needed for something to be a sport unless that diversion; recreation or pleasant pastime is a competition with rules.. Even shooting marbles is a sport, but it meets the three primary requirements
There is a physical activity.
There are rules.
It is a competition.
Simply riding a horse around, regardless of what you might be doing or how challenging it might be, is not a competition with rules you must follow. Any more than jogging to the gym or driving a car to the store or building a fire in the fireplace. However, building a fire can, by definition be a "sport" if you set rules for how it's to be done and have two or more people compete (it's not likely to become a recognized sport, but it would still meet the basic requirements)
If someone knows of a recognized sport that does not have rules and involve some form of competition I'd love to hear about.