I do know that my IP address is private, i.e. not fixed but privately assigned, as would be the case for most of the world.
Here is an example of the FLAGS, as seen by Fiddler, used when communicating with ArcGIS Online:
SESSION STATE: Done.
Response Entity Size: 8482 bytes.
== FLAGS ==================
BitFlags: [ClientPipeReused, ServerPipeReused, LoadedFromSAZ] 0x218
X-EGRESSPORT: 50266
X-RESPONSEBODYTRANSFERLENGTH: 8482
X-CLIENTPORT: 50265
X-CLIENTIP: 127.0.0.1
X-SERVERSOCKET: REUSE ServerPipe#6
X-HOSTIP: 50.19.238.218
X-PROCESSINFO: arcmap:9028
UI-COMMENTS: [#10]
I believe that this says that my address is 127.0.0.1 (which I believe is localhost/loopback) and that the server address is 50.19.238.218.
requestip should get my IP address, which is typically not very unique because of the above. What I don't understand is why one server would see one thing and another see something different. Perhaps ESRI is also looking at other information if the address is the loopback address?