Dozens of rig types have been identified in Austronesian maritime cultures (see this post for an illustration showing a large selection) -- so many, in fact, that it led to confusion and dispute about which types were related to, or ancestral to, others. And since anthropologists often view the diffusion of technology as a means of analyzing cultural origins, that confusion served as both cause and effect for disputes about how the Pacific came to be populated.
In Wangka: Austronesian Canoe Origins, Edwin Dorran, Jr., took a stab at grouping the principal Austronesian sail types into nine categories. As shown below, these nine actually group into two classes: sprit rigs (A-F) and lug rigs (G, H, I). The following brief descriptions are derived from Dorran's text, using Dorran's names for each. (Dorran assigned names as a matter of convenience in the discussion; he acknowledged that many of them were known by a variety of other names.)
A. Double spritsail. Confined to a few locations in the Indian Ocean at the time of publication (1981). Both spars are sprits -- there is no mast on this rig (and arguably, none on any of the sprit rigs described). As shown in the photo below, both sprits hold up only a corner of the rectangular sail. Possibly the grandfather type of all Austronesian sprit rigs.
B. Common spritsail: rig appears with and without the boom. Essentially identical to the European sprit rig.
C., D., E.: Oceanic spritsail: D and E differ only in the angle of the after sprit, which, if horizontal enough, can be termed a boom, as in E. Type D, pictured below, was by far the most common at the time of writing.
Dorran considers Type C (pictured below) to be "an extreme form of the basic Oceanic spritsail," because it meets his definition of a spritsail, viz: "Spritsails are supported on two spars with are close together or touching at the lower end and divergent to two sail corners aloft." To me, this is putting the horse before the cart. Even if the rig shown meets the definition (debatable), I would argue that the definition is fairly meaningless if it encompasses a sail so manifestly different in shape, type, and most likely, origin. (As Dorran notes, David Lewis called this an "inverted, triangular sail" and classed it among a category of "claw" sails with "a deeply incurved free edge".)
F. Crane spritsail: The word "crane" is being used like the hoisting machine, not the bird. Dorran argues persuasively that this is not a variant of the lateen sail. Where all the other rigs shown here are "tacking" rigs, this one is the famous Pacific "shunting" rig. The additional spar pivots fore-and-aft at the base, so that the entire rig can switch ends in the boat, allowing the canoe to switch directions relative to the wind while always keeping its (single) outrigger float to windward. In the photo below, there is a platform on the boat's lee side, on which crew (in this case) or cargo can be placed as a counterweight against the outrigger.
G. Rectangular boom lugsail.
H. Square boom lugsail
I Trapezial boom lugsail
Dorran believes these three are closely related, and that the different sail shapes represent a fairly minor distinction. He includes the term "boom" in the name to distinguish them from lugs without booms and from those with sail battens (i.e., Chinese junk-style sails).
All illustrations in this post are from Dorran, except of the double sprit rig photo, provided by boat designer Michael Schacht, editor of the blog Proafile, about all things proa. Many thanks Michael!