Words can have different meanings. The term "flat bow" originally was used to differentiate bows with wide and basically parallel back and belly from the narrow, stacked limb of the longbow or English longbow. This became useful during the 1920s/1930s as shorter, wider bow limbs became popular.
Prior to the introduction of fiberglass, the terms "longbow," "long bow" and "English longbow" were inter-changeable.
It was only a short time that these shorter, wide limbed bows came to take on a consistent, repeatable design; wide, flat, thin limbs with a narrow, stacked [deep] handle. Ben Pearson made thousands of these typical flat bows from hickory, as did several other companies. At this point "flat bow" took on an added definition, one that defined this “specific” bow design/type.
Both “flat bows” and “long bows,” are by definition, straight end bows. Straight end bows might be “setback” or “reflexed” from the handle and might have a minor curve at the limb end, but never “intentionally” deflexed.
As better materials became available for use in bows the “Golden Age” of bowery ushered in nearly unlimited design opportunities….and boy did we take to it.