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A starter is basically a 'nice name' for a capacitor, just like ballast is a 'nice name' for an inductor.
This is because the ballast used in fluorescent lamp is inductive. Therefore this starter (capacitor) is to counter the ballast in order to correct the power factor.
Try to recall the power triangle of Real, Reactive and Apparent. Inductive + Capacitive corrects the Apparent as close as possible to Real to give close to Unity.
I don't agree with you, when the starter contacts open the ballast (which is a coil) gives a high voltage spike so that the fluorescent lamp lights up, I don't think that a capacitor in place of the starter will turn on the lamp.