The Western draw is the weakest, and fairly easy to learn. It's useless on horseback.
The Turko-Mongol style is to use the thumb or thumb ring and a much stronger bow. More power and accuracy, easy to use on horseback.
So that's two styles, and they can fairly easily be separated based on Self v. Compound bows, as one only sees the full benefits of the thumb draw on a much stronger bow.
But then we get to the Persians and some of the other Aryan tribes related to them. They use a draw that allows them to hold multiple arrows. It's a stronger draw than the Western style, and allows one to fire very rapidly.
I can't think of any convenient way of differentiating this from the other two, perhaps making the 'Persian Bow' its own skill and ordinary thumb drawing simply compound bows? Persians also used double curved compound bows from their steppe origins, but elements of their military trained for massive volley firs rather than targeted shooting ala the Mongols. Any archer can shoot for volleys, but not as quickly as with the Persian style. Yet Persians would possibly also know the thumb draw, as they would probably use more conventional archery when hunting (no need to make the deer a pincushion).
So, basically, my issue is that it's not properly a separate skill but an important practice or technique certain people practiced. It is militarily important, as heavily armored wings of archers on horse were a big part of Persian battle control.