Yes, only eHydro equipped machines have the extra cooler in front of the radiator. The cooler is Deere part number AM122482, manufactured for Deere by Modine.Do only eHydros have the cooler??
The corrosion in the pictures was absolutely caused by the battery based on my multiple experiences with the same problem. The problem only goes away when the stock battery is replaced by a sealed "gel" type battery. Deere provided me with a sealed Interstate battery on the tractor's second trip to the dealer for a corroded cooler and battery tray.
4000TEN owners still running the stock battery would be wise to swap to a sealed battery as soon as possible; the new hydro cooler retails for around $180.