I honestly am trying to understand the point being made. Isn't when the day starts obvious in either case? We know when the new _day_ starts because the date rolls over at that time. No one thinks the day starts at 12:00 in the 24-hour system, or 12:00pm / noon in the 12-hour system?
> ... there's a single point where the new day starts ...
Is this what you're saying: that in the 24 hour system, there's one point in time where the clock resets from a high number to zero, and that's also when the day starts. Therefore, it is "not confusing."
I felt rather that the problem the OP stated was simply "I am used to the 24 hour system and I was communicating with someone who was used to the 12 hour system. They naturally gave me a time of 12:00am but I didn't notice that am part and assumed they meant the same "12" that I understand as "12" in my system. So I missed an exam."