Must have something to do with scripting/coding and the fact that you're not playing with a Greek faction. Usually I have a revolt from time to time that needs to be put down and I tend to sack or even enslave a settlement to restore order. Those cities need a recruited allied governor right away if you can hold onto them long enough to install an allied government (the one just before choosing an oligarchy or democracy). You can also improve public order by putting in better generals as governors right away, and by that I mean ones with traits and ancillaries that specifically affect public order. Generals with lots of "influence" (i.e. the status bar with green laurel wreaths) convert culture much quicker than others and have a public order bonus. The problem, it seems, is that your general "Antipatros Kianos" sucks as an administrator. Surely you have a better general/family member for this? I mean, not only is his command star status at 3 (an okay field general at best), his influence is only at 1. That's not good.
Also, your second picture of Corinth no longer had that many soldiers garrisoned there. That's another problem. In addition to a good governor, you're going to need lots of troops for this, with fresh troops to replace the damaged ones, I'm afraid.
If it's driving you insane, you could always cheat, too.
Code:
add_population Korinthos -4000
add_population Korinthos -4000
give_trait "Antipatros Kianos" "PeopleHappy" 3
give_trait "Antipatros Kianos" "GoodBuilder" 3
give_trait "Antipatros Kianos" "GoodAdministrator" 3
give_trait "Antipatros Kianos" "KindRuler" 3
give_trait "Antipatros Kianos" "Just" 3
give_trait "Antipatros Kianos" "Pious" 3
give_ancillary "Antipatros Kianos" "judge"
give_ancillary "Antipatros Kianos" "playwright"
give_ancillary "Antipatros Kianos" "historian"
give_ancillary "Antipatros Kianos" "rhetor_hellenikos"