That's an additional reasoning you have to do in order to explain the non-intuitive fact that the new day starts when transitioning between hour 11 and 12. You can't come up with that conclusion based on clock system alone, you need to reach for the calendar. It makes you jump additional mental hoops, making it error prone (as evidenced by this thread).
> but I didn't notice that am part and assumed they meant the same "12" that I understand as "12" in my system
No, people who use 24 hour system are usually familiar with 12 hour system as well (guess what's there on analog clocks?). It's just this one special case of 12 AM/PM that's confusing, because you can make convincing arguments for having it both ways.