I was able to duplicate the behavior that you all are talking about...which is good and bad. Good that I can duplicate it, bad because it means a problem exists.
I am also unable to delete appointments created in lightning on Zimbra. Since Lightning only reads what is present in Zimbra...we can conclude that Lightning is unable to do one of two things:
1) Lightning does not know how to communicate with Zimbra properly or Zimbra does not comply with calendar standards in this instance and Lightning is updating according to standard but Zimbra is ignoring it.
2) Zimbra has the wrong permissions
To resolve this, I attempted to setup my secondary email with full share/manage access to the calendar. I then used the same URL noted in my how-to to plug in my Zimbra work calendar at home. I attempted to delete an example appointment via Lightning and found that this could still not be done.
Therefore, I can conclude that this is #1 from above. Which one it is, I do not know...we'll need input from Zimbra on whether or not standards are followed in this instance and then we'll have to check with lightning to see if they also follow standards (mozilla is usually pretty good about following these things though).