What is the purpose of this coil in a medical defibrillator?
The inductor is there to shape the waveform.
Depending on the values of the capacitor and inductor, the energy from the capacitor will be delivered in a pulse of sine waves around 10 milliseconds long.
The diagram you have is fairly primitive, and dangerous besides - one side of it is still connected to the AC source when defibrillating.
That method of generating and shaping the pulse (as well as the shape of the pulse) also appear to be outdated.  The Wikipedia page refers to a newer method (biphasic pulses) that has been in use since the 1980s.
To top it all off, more modern defibrillators use more advanced techniques to generate the pulses so as to deliver enough energy to the heart to stop it (yes, stop it) without causing excessive collateral damage (burns and such.)

The problem that defibrillators &quot;fix&quot; isn't a stopped heart, but a heart whose various muscle sections have gotten out of step.
Rather than beating in step, the muscles are all twitching at various rates.  The defibrillator smacks them all and makes them stop.
Thereafter, they start beating again - with a bit of luck, they start up synchronized and stay that way.  If not, zap 'em again and hope for the best.