The physical restraint used in ponyplay is the primary focus of the bondage pony. During this type of ponyplay, heavy bondage will be used to contain and restrain the pony. Indeed, in some cases, it may not be possible for the bondage pony to do anything due to the extent of the bondage.
A bondage pony should not be confused with a bondage horse; the latter being a device in which a person is forced to straddle a wooden board like they would sit astride a horse. This causes all or part of the persons weight (depending on how they are restrained) to press the genitals onto the small hard surface of the board, which obviously causes great pain.
Many of us engaging in ponyplay enjoy the bondage (as well as the D/s and SM) aspects of this fetish. Moreover, there are times when heavier bondage is used on any type of pony (e.g. during breeding, for punishment, for veterinary procedures, etc.). However, most ponies also perform other activities such as training, and grooming that necessitate the use of less restrictive tack. While bondage ponies may still participate in these activities, the degree and extent of bondage will usually take precedence over other factors.
Another classic bondaage pony position is the hogtied bondage pony, where the pony's hoof boots are bound to the pony's hoof mitts (hands). This is quite effective and aesthetically pleasing and many bondage ponies will enjoy the restriction provided.
However, you can make the bondage positions even more extreme in cases where the pony needs to be punished, or where immobilization is paramount (for example a veterinary procedure that the pony would strenuously object to). One such variation on the classic hogtie is to bind your pony's elbows tightly together so they touch, then instead of tying the hogtie rope from the hoof boots to the hoof mitts, you tie the hogtie rope from the bound hoof boots to the pony's bound elbows and and watch him try to struggle in futility. However, this position requires very careful attention on the part of the handler, since the position is extreme and can easily cause loss of circulation and place immense pressure on the pony's diaphragm:
Alternatively, you can tie the pony's hoof boots to the pony's bit (ideally a harsh bit as shown in the linked video, where the bit has a long tongue port, which is pulled ever depper into the pony's mouth as he struggles, causing his gag reflex to get stimulated, and thus discouraging further struggling/movement):
There is no clear cut distinction between a bondage pony and a pony that likes bondage, but the classification is made more obvious at the extremes (for exmaple in the previous two photos, there is little the pony can do - this can be dome as punishment or to show him off to others, or for your own amusement at his struggles : ).
A pony can be purely a bondage pony or may be a bondage pony and another type of pony (e.g. a cart pony may also be a bondage pony on occasion), or just a pony who enjoys (or whose trainer enjoys) bondage