Does less settling time means that the response dies out quickly? If so the settling time for a just correct Resistance such that the circuit is slightly underdamped is least, then it's most damped(considering the definition of damped losing energy gradually, what it intuitively means). Then shouldn't it be called overdamped instead? Or maybe I'm confused with the relation between damping and settling time?
Settling time means the time it takes to get to and stay within some distance from the final value. It does not mean how long it takes for the system to settle out completely, because (for linear systems, at least) the response always gets ever closer, but never actually reaches the final value. (In practice things settle out to being indistinguishable to the final value -- but that's not what theory says).
An underdamped system rings -- it has a (hopefully!) decaying sinusoidal component to its response. As with settling, this component rings forever in theory.
There's a trade-off between settling time, overshoot, and damping. An underdamped system (usually) overshoots, but a mild amount of underdamping will often make the system settle faster, usually as long as the overshoot is less than the settling time limit. A severely underdamped system will have a long settling time, because it'll be ringing like mad.