Alright. Although I didn't figure out why MunkiWebAdmin didn't update the name of my client automatically, I did find my blind spot and figured the larger question out. My apologies for not catching this obvious thing sooner. However, in case there is someone new (like me) that didn't catch this: Changing the name of the client or deleting a client is found in the
Admin menu item at the top right in the solid black bar. From within that section you can make manual changes to the client's information including deleting them or, as in my case, change the name of it.
I can add that I did still expect it to update the name of the client as MunkiReport did, but I can live with this as a basic work around. Further, I expect deleting and letting it report afresh will update the information (although I have yet to try).
Thanks for allowing me to post!