Man I think the AE110 corollas have it too!
And yes it would come on during start up as it does the usual system checks, but I dunno why it didn't light up when you got the flat... It doesn't come on immediately though..
I think maybe, it would take a lil while for it to respond, maybe a couple hundred metres? I think the system would not automatically send off the warning light on the first check of the system, cause after all, it IS a somewhat complex situation, given bumps, potholes, etc on the roads.
Maybe it needs a few cycles to confirm a problem?