What does each node in a doubly linked list contain, beyond its value?

Every node in a doubly linked list stores its value plus two links: one forward to the next node, one backward to the previous. NIST's dictionary defines the structure as a linked list variant in which each item has a link to the previous item as well as the next. That backward link is what buys you constant-time removal. Holding a reference to one node, you can reach both neighbours and rewire them without walking anywhere.

A singly linked node knows only what follows it, so deleting it means scanning from the head to find its predecessor. That is the reason an LRU cache pairs a hash map with a doubly linked list: the map hands you the node, and the two links move it to the front.

The wrong answers each break the structure. A node with no pointers is not in a list at all; the links are the list. Pointers to every other node would cost O(n) space per node and O(n) fix-up on every insert. Storing a copy of the whole list inside each node duplicates the data the list exists to hold once.