I'm new here and I'm not FDNY, but I'll take a stab at it! (Nothing ventured, nothing gained).
Reported structure fires, for the most part, get 3 engines, 2 ladders, and a battalion chief. Upon arrival if fire is blowing out the windows, a 10-75 will be transmitted and then a 4th engine, rescue, squad, another battalion chief, a ladder assigned as FAST, the division chief and a RAC will be sent. If all units (or at least 5 units) are committed and are in use, the All-Hands is transmitted. This used to bring additional resources to the scene, but no longer does. However, this is useful information for the dispatchers, as now they may make a more comprehensive coverage assignment plan, and for any units due on the 2nd Alarm, as their ears may perk up.
The true definition of All-Hands is, at minimum, 5 major pieces are used. Getting back to the All-Hands, Under Control, this would most likely mean that the 3 engines, 2 ladders and the battalion chief assigned to the reported structure fire arrived, had a manageable fire and were able to control it with the 3 engines and 2 ladders (5 pieces). If not offered by the battalion, the dispatcher will usually quire if EMS is needed.
A fire that was extinguished using 2 engines, 2 ladders and the 3rd engine standing by would not be an All-Hands/Under Control because 5 major units were not put to work.