Rods are the red-headed stepchild of magic items. Not as prestigious as wands or staves, but a step above wondrous or "miscellaneous" magic items, they'e never had a well understood place in the pantheon of artifacts. So what exactly is a magical rod?
There are two keys to identifying and understanding rods. First, they are usable by all classes. If any class is restricted in their use of rods, it is in fact likely to be magic-users and their ilk.
Second, rods transform. A rod with one shape and one function is a magical stick. Almost every rod transforms itself; many transform the wielder as well.
Rods are a symbol of rulership and supremacy. Sceptres and crosiers are both forms of the rod. Many rods evoke this symbolism in their powers, whether conjuring rich clothing or followers, forcing obedience on those around the rod, or enforcing certain behaviors. Fighters, clerics, and to a lesser extent druids favor rods that create leadership effects. These rods are never subtle.
Rods are utilitarian. On the flip side of the symbolism is the utility of the rod. Fighters and thieves both enjoy more functional, less flamboyant rods that hold a variety of effects and tools. As a corollary to this, most rods are also weapons.
Rods are not defined by spells. Rods are not collections of spells. If an item can be defined by the spells it casts, it's almost certainly not a rod. Spells are the hallmark of wands and staffs, not rods.
So a magical rod is an item that (primarily) benefits fighters, clerics, and thieves; has multiple functions and usually changes shape; and whose most prominent abilities are not spells.
(FWIW, a lot of 1e & 2e wands and staffs are now rods by my definition. Not all, but a lot of them.)