@Anonymous You have to think of them as nights rather than days. I know, it's confusing. So in this case, you'd want to book the 12th and the 13th. You would check-in on the 12th, at whatever the host's check-in time is, and you would check out on the 14th at or before whatever the host's check-out time is.
I didn't know that on the guest's end you actually enter check-in and check-out dates, that's weird and would certainly lead a guest to think they wouldn't be charged for 3 days. I figured guests just enter the days they actually will be sleeping there. On the host's end, if you wanted the 12th and 13th, it does appear as the 12th-14th, but we know that that means you're checking out on the 14th, not sleeping there that night.
You should contact Airbnb- you certainly shouldn't be being charged for the night of day you entered as check-out.