I have never created a calendar named "private", but on iCloud, I have one. This one pops up if I create a new event. In most cases, I am able to change it to the correct calendar listed under "My iCloud Account" . The window that pops up contains two lists named "My iCloud Account" and "iCloud". They contain different calendars.
In some cases, no idea why, a new event is only listed to "private" on the secons list. And there is no option to change it.