I hope this will help make conversations about your horses hooves more effective. Some books do not go into great detail of farrier terminology as much of this was news to me as a horseperson entering farrier school. These are terms I learned which seem to be commonly accepted.
Hoof Wall: This is the most obvious and largest external structure. It is produced at the coronary band and grows distally, meaning towards the ground. The farrier mostly trims hoof wall hopefully and if shod, it is what the shoe is secured to.
The majority of the horses, shod or barefoot, weight is supported by the hoof wall. It is strong, durable yet surprisingly flexible in order to accommodate for the constant expansion and contraction involved in locomotion.
Heel: I consider the heel as an extension of the hoof wall. The wall actually bends inwards sharply next to each side of the frog. Those sharp “V” corners form heels. You can be more specific if you are referring to one or the other by using the term medial, lateral or inside, outside. Left and right become too complicated when dealing with horse feet.
Bars: the bars are a continuation of the hoof wall that protrude from the heels towards, while becoming less prominent as they near the tip of the frog.
The tapered bars become increasingly more weight bearing as the foot is loaded. As the foot expands the bars are pushed towards the ground to aid in shock absorption.
Sole: this is the horn that covers the majority of the ground bearing surface of the foot. It is constantly produced and within the hoof capsule and self exfoliating. The ideal sole is concave and thick enough to protect the coffin bone, blood supply and other sensitive structures just millimeters above.
The sole is not considered load bearing, although in horses that are kept barefoot the sole with thicken to the extent that there may be ground contact.
White Line: a bit of a misnomer, the white line is a yellowish , insensitive substance that is sandwiched between the hoof wall; including heels and bars, and the sole. It may be difficult to see the white line unless the foot is clean and recently trimmed.
The white line is produced internally peripheral edge of the coffin bone and lies beneath the laminae and fills the void that would be left between the sole and hoof wall. You could think of it as a border and bond between the sole and wall.
Nails are driven into the white line ideally and exit via the hoof wall.
The white line is a load bearing structure except for the area called…
Seat of Corn: sometimes called the seat of heel. This is an area not a structure. This is the point of the foot where the hoof wall folds back on itself. That juncture is especially prone to irritation in some modern horses and a lameness caused by this is called a corn.
Frog: this is the triangle pad of moist and pliable horn situated between the heels. The frog aids in traction, shock absorption and circulation.
The frog, like the other structures are continually produced by a corresponding sensitive structure .
The frog is a weight bearing structure.
Commisures or Medial and Lateral Frog Sulci: these are the two groves formed by the joining of the frog and the bar. This is the “V” shaped area that is cleaned with a hoof pick.
Frog cleft or Central Sulci of the Frog: This is the grove that is made between the left and the right side of the frog, Some horses are prone to getting a deep thrush here that can go undetected.
I hope this helps you say what you mean, hopefully I will be adding to this in the very near future.