I agree that D&D has never gotten fey quite right.
In folklore and mythology, I think it is easy to label fey as either a sub-set of the general idea of "spirits", or as beings which exist on the boundry between being spirits and being living creatures. They are mysterious and otherworldly, superhuman and supernatural. They exist to punish humans who dare to intrude into their realms, or to trick human into coming into their realms so thet the human could be toyed with. 99% of the time, a human who even meets a fey endures misfortune, though the degree differs. They could be beautiful, childlike, or monsterous. They are the classic butterfly winged faeries, elves, and trolls, all wrapped up in one mix.
The only part of D&D I have seen yet to get faeries even close to right are the Daughters of Sora Kell in Eberron, but they arn't fey in that setting.
Hayao Miyazaki's Spirited Away is a great example of a classic fey story. The old witch of the forest, the Baba Yaga, the witch from Hansel and Gretel, and the witch from Spirited Away, all have much in common with the dark side of the fey archetype.
If you have played Legend of Zelda: Twilight Princess, the character of Midna, especially early on, is very much a fey character.
I think it is important to remember that the most common fey story of all time is the story of the person who accidentally wanders into the world of the faeries, spends some time partying there, and then leaves, only to realize that decades have passed.
Also, Chinese and Japanese animal spirits, such as the Kitsune or Tengu, have more in common with Fey than Greek Nymphs, including Dryads and Nereids.