Marty,
Is correct. In fact the way the speaker mechanical and air suspension is designed is very important as to how it will sound. (actually far far more important than amplifier damping!!)
Here is a third term: "critically damped". This has been generally accepted as the best sounding most practical approach. This is neither overdamped or underdamped - it means that when power is removed then the speaker goes to the rest position faster than any other design WITHOUT additional oscillations.
Here is a link to
Damping. The red curve is overdamped (like a screen door closing - uncommon in speakers - needs lots of force to make it move). The Blue curve is underdamped (quite common in speakers with a bass boost/hump). The Green curve is the "goldilocks" curve - critically damped or "just right" (often regarded as the optimum setup for a woofer)
You may ask - Why then do speaker designers use anything else except "critical damping"? Basically it boils down to tweaking to get more output efficiency or more bass extension from small boxes - for example an underdamped driver with a port can get you more bass (higher efficiency) but the bass may sound muddy or warm (and won't be very smooth in freq response). Conversely, a critically damped sealed box design will sound tight and punchy in comparison but won't have anywhere near the bass extension and efficiency. An overdamped design will sound very tight but will be extremely inefficient - needs lots of power - high power Class D SS amps and Active designs may make this a more viable option in future.
The above may explain why some ported small subwoofers with high bass efficiency may sound impressive but muddy or non-musical.