The old Giver is no longer receiving memories from anyone, so technically he is no longer a Receiver of Memory. In fact, he probably hasn't been for a long time, as the one who trained him must be long dead. But now that Jonas's title is the Receiver of Memory as well, it only makes sense to have a new name for the Giver to acknowledge the change in his role. 

The relationship between the two exposes a weakness in the community, and that is that there is only one Receiver of Memory at a time unless training is taking place. This master/apprentice relationship takes on new significance when we realize that outside of the Giver and Jonas, there is no one else in the community capable of holding these memories. So, as the training continues, and we see how vulnerable the community is, it is interesting to think about the Giver's new role as one that, perhaps, should have taken place long before him needing a replacement. What if the Giver had died of natural causes prior to training a new Receiver?