Here's an 1805 fashion plate from The Lady's Magazine for "London Fashionable Full Dress" just so that you know to what I am referring.

I don't believe that feathers in full dress implied that the wearer had been presented at Court. Upstanding plumes were a distinctive feature of proper court dress in Britain during JAusten's time, and had been for nearly 75 years. Court dress really was a fossilized form of attire.

(The distinctive feature of French court dress, for example, was lace lappets hanging down either side of the woman's face.)

They just plain old liked to wear feathers when they dressed bang up to the nines, from any bird that took their fancy: egret, parrot, ostrich, peacock, pheasant, bird of paradise, etc.

Re the Prince Regent's Plume - it became trendy to show support for the Prince of Wales in the Regency Debate by wearing the distinctive insignia of the Prince of Wales: three upstanding plumes contained in a crown. Sometimes it was embroidered onto a dress, or formed into a brooch. But I don't think the Bingley sisters are involving themselves in that particular political debate by wearing their feather head-dresses, at least not in 1813 where A&E set their particular screen version.