You have good points, but you're splitting the wrong hairs -- the difference between a broom and a brush is not the length of the bristles or the shape of the object, but rather how it's used, as kentix and the others have said above.However, in western countries, original brooms have disapeared (they remain only in children books: Witch broom.)
Modern brooms are made of large brushes with long handle.
In other words, a modern broom is a brush with a long handle as clearly stated in many English dictionaries.
"Fuller Brush" is the brand. They sell brooms, brushes, dusters, mops, cleaning solutions, and even hydro-jetting sprayers. The links that you gave are to brooms and broom heads, not to brushes or brush heads; please don't be confused by the presence of "brush" in the brand name into thinking that brooms are brushes. In fact, that company became famous for its hair brushes. Fuller Brush Company - WikipediaFuller Brush
The link you provided to the Yocada product says "broom brush" in the title, but that product is a broom. Keep in mind that on the internet, the more keywords you have in the title, the more frequently your product will appear in searches. So the web designers added related keywords to the title of the product, resulting in a Frankenstein-like product name:
Yocada Push Broom Brush Stiff Bristles Broom Head Telescopic Heavy-Duty Outdoor Commercial for Cleaning Bathroom Kitchen Patio Garage Deck Concrete Wood Stone Tile Floor