Any changes that need to be made to an appointment from Acuity need to be made within Acuity. For instance, if a change was made to an Acuity appointment in Google calendar, then that appointment change wouldn't be reflected in Acuity, and notifications wouldn't be sent out to the client. So in order for an Acuity appointment to be edited, it needs to be edited inside of Acuity.
Edits to events from other calendar programs need to be made in the calendar program the event was initially made in as well. Any changes to Google events need to be made in Google, any changes to iCal events need to be made in iCal, and any changes to Outlook events need to be made in Outlook, etc. All changes must be made in the program they came from.