Your reference seems to relate to SIP transfers. Their problem seems to be that it is working the way you want! (I didn’t read it in depth, but it is possible that Asterisk is being bypassed.)
If you don’t know whether you have a SIP or features transfer, ask the person who configured Asterisk and the phone.
Normally if the word Transfer appears on the phone’s UI, you have a SIP transfer, in which case, if it is actually an attended transfer, the call to C is made, and therefore the initial CLID is set before Asterisk has any idea that a transfer is coming up. You need the 1.8 changes, and a phone that supports them, for there to be any possibility of the CLID being updated when the transfer is completed.
As noted before, some phones make an attended transfer, even if their user interface indicates a blind one. An actual blink one should show party A’s CLID, as Asterisk does know that it is a transer when C is first called.