ACARS and Pings
IMO, this is how the protocol works.
ACARS <-> A/C Terminal <-> INMARSAT Satellite <-> INMARSAT Ground Station
Typically messages go between ACARS and the INMARSAT ground station (GS) and, even though the messages pass through the the aircraft terminal and the satellite, these are invisible, just like when you talk on cell phone, the terminal in you phone, the tower and network are (hopefully) invisible to you.
When ACARS is off or when the aircraft doesn't have anything to send for a long period, the aircraft terminal and the ground station stay in contact so that the INMARSAT system can keep track of who is in its communication environment and who is not. This is done by the the GS querying the terminal once an hour. The GS sends a message to the INMARSAT satellite, which generates a "ping" to the terminal. If the terminal can hear the satellite, it responds with a simple message saying yes, I am still here. If there is no reply, then the INMARSAT GS probably eventually logs the terminal off and stops polling - we have not been made privilege to this - just speculating on this, having developed other satellite protocols. This is also why you shouldn't carry your cell phone in your pocket, right next to your jewels.
Now, the terminal also may have a timer which says if I don't hear from the satellite in a certain time (longer than the 1 hour above) then it may send a message to the satellite saying hey, I am still here. If it doesn't get a reply, then it possibly tries to connect to other INMARSAT satellites. We haven't been made privi to this part of the protocol either.
If the terminal is powered up or reset, it would probably try to initiate a connection to an INMARSAT satellite / GS. IMO, this is what I think the final ping is. If it was, did the GS / satellite try to handshake back with the terminal - if it did, then maybe it is possible to determine the final location and doppler.