No nonsense at all. I read all three paragraphs. Nowhere did it mention or describe a baseball type cap. It described various styles of HATS. Another pet peeve is the lazy, or careless reference to T-shirts. Back in 1947 when that style came on the scene, they were Tee Shirts; alluding to golfing.
The definitions do not mention or describes any style of hat. They give only general elements required to fall within the definition of hat. A baseball type cap falls within those definitions and is excluded by none of them.
How is a baseball type cap not "a covering for the head, especially one with a shaped crown and brim." ["google images BRIM"]
How is a baseball type cap not "a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory?"
How is a baseball type cap not "a shaped covering for the head worn for warmth, as a fashion item, or as part of a uniform?"