In theory the "signal" stops and starts as recorded so that is accurate. The problem is the woofer does not stop right away. In a lower damping factor amp the output resistance will help dissipate the energy in the woofer which can give it a more controlled stop.This is of course the theory but in practice (IOW the real world) is not a thing. The woofer never 'stops'; its always in motion; there is no recording where the woofer 'stops'. The only way for it to 'stop' as often described abover is if a DC pulse is being reproduced, which is something that neither the amp or speaker will ever have to do. On this account I've often viewed this as a red herring.
More to the point, the idea is that the woofer will continue to oscillate after the initial pulse. But in reality after the woofer moves the first way, the audio signal directs it in the opposite way; it never gets the signal removed and so can't ring. This is why amps with a low damping factor can do quite well in the bass. The real problem isn't ringing or distortion, but getting too much bass if the amp fails to reduce power into impedance peaks of the woofer's impedance curve. This isn't a control issue (which is also a problematic idea), its simply that an amp with insufficient low impedance will simply make too much power.