From photographs I found online, it appears that an American Flatbow is not flat. It's curved. And some of them are laminated. And it's thicker where you hold it in or about the middle than it is at both ends.
So how does that differ from the other types of bows?
Lots of different terminology, not always used consistently by even those with experience. What you seem to be looking at is the difference between an English longbow with it's "D" cross-section for the limbs and more modern longbows, often in America, with the same rectangular cross section for the limbs as in recurves. However, both are longbows because, when strung, they form a single curve with the string being the straight portion of a "D" shape. Also, the string loop touches only the tip of the bow. Unstrung, a "longbow" may curve, sometimes in multiple directions if it is a "reflex/deflex" form. My understanding is this form helps reduce hand shock.
A re-curve, on the other hand, even when strung, curves toward the archer, but then "recurves" away with the tips ending up close to parallel to the riser at full draw.
Here are some articles I found that seem to match my understanding.
Types of Traditional Bows
American Flatbow vs. English Longbow: Which Is Better?
Flatbow (Wikipedia)
Most flat bows start was a flat piece of wood and are mostly flat when not under tension, which happens when you string it. Most also have a flat inside surface. The other major variant in use when the generic name 'flatbow' was first used was the recurved bow that look like snake slithering along the ground with 3 obvious curves in it or the very curved bows short bows used by some of the American Indians. Today you all sorts of things with the compound bows and other fancy stuff.
In general, a flatbow and a longbow are the same thing and are made from many different woods. What many people think of and call and English Longbow is, in reality, a Welsh Longbow and it was very different from every other longbow or flatbow in use at that time as it was the only one made from the Yew tree with some heartwood and outer-wood as a natural laminated wood. They got a huge reputation through the many Welsh longbowmen who were employed by the English against the French. The Yew longbow had more than double the power of the usual longbows in use by everyone else.
Most flat bows start was a flat piece of wood
You can get modern flat bows that are made from fiberglass or other composite materials.