Awave and a pendulum are both oscillators. why isn't a pendulum a wave?
..

-- A wave travels from place to place. 
   A pendulum just sits there.

-- A wave carries energy.  A pendulum has energy,
   but doesn't take it anywhere.

-- A wave is a traveling pattern of maximums and minimums
   that all exist at the same time but in different places.
   A pendulum is a stationary pattern of maximums and minimums
   that all exist in the same place but at different times.
   (I made this one up just now;  I really like it.)   

What makes you think that all oscillators must be pendulums,
or must be waves, or must be both, anyway ?

A guitar string is an oscillator that has waves on it,
but no pendulum anywhere nearby.

A car horn is an oscillator that sends waves out
but there's no pendulum anywhere nearby.

An analog wrist watch has an oscillator in it, which is
actually a torsion pendulum, but there are no waves
anywhere nearby.


Going just one step further with the logic of your question ...
If the logic held water, then we would have to ask:
A horse and a dog are both animals. 
So why isn't a horse a dog ?