Jim,
You were using the quote to back up your claim that passports are not currently required between UK & Ireland (North or South)and quoting a reference to the Schengen rules does nothing to support your case.
The current freedom to move between these areas is because there is a "Common travel area" covering UK, Northern Ireland, Republic of Ireland, Isle of man and the Channel Islands. This 'common travel area' is nothing to do with the EU.
UK & Ireland (both parts) are outside the Schengen area. The Channel Islands are even outside the EU.