The other day, this question (Why did wetting down canvas sails increase the speed of a sailing ship?) was asked regarding the historical practice of wetting down sails. The current answer quotes several sources describing two factors. First, is to swell the fibers of the sail and to seal the spaces between threads so that the wind blows along the sail, not through it. This makes perfect sense to me. The second, per the quoted sources, is to "to flatten their canvas" or "likewise making the sails stand flatter." This lies almost in direct contrast to my intuition.
I've been taught (and I admit I am barely a novice sailor, and mostly a power-boater or paddle-boater) that a sail acts similar to an aircraft wing, so to make it create aerodynamic lift, it needs to have a wing-like shape to it, called a belly. This article describes the concept and why it is important. At first glance it would seem that flattening a sail would reduce the pressure differential between the windward and leeward sides, thus reducing the lift generated by the sail, taking speed away from the sailboat. Yet both quoted sources in the linked answer refer to either racing or pursuit, and the wettened flat sails aiding the vessel's speed.
In trying to justify this, I wondered whether it mattered if one was dealing with a square-rigged sail versus a fore-and-aft sail, and also if the vessel is on a beat, reach or run. For example, the sail on a square-rigger on either a broad reach or a run is essentially using the wind like a pushing hand, so maximizing area of resistance is more important than creating a lifting surface.
One of the sources tends to confirm this partially:
general practice ... to cut and sew the sails so they would form themselves into a bag to hold the wind, although the advantage of flatness of surface for plying to windward was well understood.
At the same time, it contradicts my thoughts above, as "plying to windward" is equivalent to beating, and the reference to yachting and the America suggests fore-and-aft rigs.
I just can't wrap my brain around wanting flatness in this situation. Please help me understand the interplay between the various forces acting on a sail and how the amount of belly helps and hurts in various points of sail.