Correction of Diploma Having Two Types of Honors
In my opinion, the most important statement in your question is "I ... am now a Ph.D student."  Assuming that you follow through in your doctoral program, everything about your undergraduate career will rapidly fade into insignificance.  Please understand that I do not mean to downplay your accomplishments in any way: it is simply that you are entering a new realm in which you will be judged very differently on a very different type of accomplishment.

Given that, what's important here is to make sure that nothing on your C.V. could be interpreted as a lie in the unlikely case that somebody bothers to check with your undergraduate institution.  Check what the CS department believes about you, and write that down on your C.V.  You can always update it later if the bureaucracy gets fixed.

Beyond that, it's really more for your own personal satisfaction than for any career value, particularly as regards the pieces of paper that you were shipped.  My own personal experience as a working scientist... I recently rediscovered my degrees while going through a box throwing out old papers, and they have now been relocated from a box in a closet to a box in the attic.