Basic Balance Position - a two-point or three-point for the rider?
Most of the time we ride in a three-point position. This is sitting on both seat bones with the tail bone in contact with the saddle too. The imagery of a three prong plug as our seat being plugged into an outlet, the saddle, may help. There are times that we should use a two-point position to allow our mounts to balance underneath us easier than if we maintained a three-point contact. By allowing both seat bones to be the point of contact and closing our hip angle slightly to lighten our weight in the saddle the horse, mule or pony can be freed up to use its hind legs while we stay balanced so as not to interfere.
You should get into two-point when there is an obstacle to navigate such as logs, rocks that may cause some slipping /sliding, a water crossing or hills. Faster gaits ridden in a two-point will keep you over the horses center of balance better. Make sure your back still stays straight as you do not want your weight to just move forward creating a burden on the forehand.
If the obstacles become larger and may require a jump from horse like a ditch or larger log you can allow your hip angle to close even more and land in your heels. Thinking of landing in your heels allows your joints to stay supple and all close helping you as the rider maintain proper balance. If you tippy-toe instead you will most typically end up in front of the horses center of balance and go for a rough ride. Another way to think of this is, "toes up". Another thing you can do to ensure you stay relaxed is to exhale as you negotiate the obstacle. This keeps you breathing. Holding your breath while you ride creates tension and soon cramps too.
If you are riding over flat ground at slow speeds sitting deep into your saddle in a three-point position is appropriate and will help you stay in the best of balance. If you ride at speed or are negotiating rough or unknown terrain a two-point is the best position for you to stay in balance and will allow for the horse to move as needed under you to be able to maintain their balance. Ride with your toes up all the time to make sure your joints are flexing and able to absorb shock.