Hi, if you look at a starter it is vertually a link of wire, there is a tiny wire, when the juice is turned on the wire heats rapidly and bends away which allows the juice to flow through. When you turn on a fluorescent tube, the starter is a closed switch. The filaments at the ends of the tube are heated by electricity, and they create a cloud of electrons inside the tube. The fluorescent starter is a time-delay switch that opens after a second or two. When it opens, the voltage across the tube allows a stream of electrons to flow across the tube and ionize the mercury vapor (I didn't write this) it takes vertually no time to do so and whilst it is doing this the tubes flickers as the juice is going through.
If the starter is faulty the wire can already be fused or melted together, but the power isn't enough immediately to strike the lamp as it is not there straight away, hence the choke, and the gas in the end of the tubes are still heating, however as the wire is already shorted power is getting to the end and when you remove the starter the action does actually spark the unit into life due to how it is explained above. When you put the starter back in the fact it is broken can inhibit the flow even though the wire is fused together.
The starter may work on other fittings if the control gear is in tip top condition, however this will be very temporary as the starter is actually knackered. Your lamp and starter were kanckered in your light, now both have been replaced it will be fine. Throw the starter in the bin.