Yes, the light should come on pre start.
A flickering or permanently on light is usually low fluid but if you are sure that this is at the correct level then maybe not.
Two things to try then, you will need some help for number 1.
1. Remove the top of the reservoir and then manually move the plunger up and down and see if the light reacts as it should, ie, OFF when the plunger is up and ON when the plunger is down. It has been known for both the plunger to stick and for the switch inside the top to become intermittent. A bit of WD 40 dribbled into the housing from the spine of the plunger should cure this.
2. The handbrake ON switch is another possibility, it happened on mine. Sometimes the light would come ON at start up, sometimes not. Operating the Handbrake gave the same results, sometimes OK, sometimes not. The problem was the handbrake switch which is located under the handbrake inside the gaiter. It's a weird looking switch but an application of switch cleaner and then several operations of the handbrake cured this. This was over two years ago and no bother since.
Off you go into the sunshine and have a look, I'm pretty sure that one of these suggestions will be the answer.