"There are three houses of Elves in Tolkien's lore: Bëor, Hador, and Haleth."

Nope. These are the 3 houses of the Edain i.e. the humans that choose to ally with the elves. The author might be thinking of Ingwë, Elwë, and Finwë who are the kings of the 3 houses of elves who choose to the Valar in Aman.