It clearly has an effect on Critical rates (and eyeballing it, about 1% critical per point of luck), but as Critical is ALSO a separate attribute (and unaffected by Luck), how exactly does Luck work and what does it affect?

Almost certain Luck also affects drop rate.

they should have explained the battle mechanics more detailed ...
at least what i know: inflicting status effects, being inflicted status effects, cards etc.

Also seems to me that is also serves as a crit protection stat, as my max luck chars seemed to get critted a lot less than the low luck ones.