Based on the left drawing: Clocks on train, Bolts simultaneous on train: the train observer of course sees the 2 bolts at the same time and the clocks tick to the same number. The train station observer see the right flash first (it travels further to reach the center of the left moving train).
So your question is working backwards: how does the platform observer see the 2 clocks agree (which he has to: if they both display 0.02 he sees a big red 0.02 on each--there is no Lorentz transform that makes a clock display change)--when he thinks they should start at different times? This is good question and the resolution to the paradox is as follows:
The 1st problem is how do the clocks start? Note that they start simultaneously with the bolts in the train's reference frame: the bolt and the timer start have a space like separation, so it's an experiment that can't be done.
Nevertheless, it's a thought experiment: suppose the clocks just happen to be started correctly on the train. What happens is the platform observer says the clocks start at the wrong times. He see the following (time ordered):
Right lightning bolt
Both clocks start from 0.00
Left lightning bolt
Bolts hit middle, both clocks stop, reading 0.02.
So for the platform observer, neither clock measured the transit time of the lightning flash it was observing.