More specifically, the first day that you enter a PARK using the AP is the first day the AP starts "counting down" (AKA, the AP's Anniversary Date.)
That day may or may not be the day that the AP is "activated" at a ticket window or Guest Relations.
(For instance you COULD "activate" the new AP at, say, Disney Springs' Guest Relations on the day you get to WDW, but not actually enter a park until a day, or so, later.)