Chiang Mai airport is the most lay back in all Thailand. I travel through that airport more then 10 times. First, you overstayed by 1 day, not 12 hours, it's counted in days. Second, overstay is 500 baht fine per day, but for a day usually you don't get charged. Do you remember if you got charged and if so, how much? Third, no, you won't have problem when you come back. The stamp you got is a "overstay notice", which tens of thousands of foreigners get every year. I have had several of these stamps in the last 12 years. It's not a problem as long as you overstayed for less then 90 days. Anyone who tell you otherwise is using google at their information.
Do not listen to the newbies below with their "2 weeks" of life experience in Thailand. You DO NOT have to pay given the situation explained above. If the hospital refuse to speak with your insurance company with phone in hand you can at any time leave the hospital. The hospital is "testing" every foreigner resilience in such situation.
Some people have money and don't want to leave the country, and on the other end some people don't have 400k for a marriage visa, etc, but still want the visa. Yes, everyone gets their cut, but it's valid and authentic, all the stamps are the same you would get if you had done it yourself.