save client-side method is the same "onBeforeSave" from the server-side logic. There is also onSaved method that is the same as "onRecordSaved" in the server logic. And the this.isAddMode() check that Vladimir mentioned works perfectly if using it both in the save and onSaved methods on the client-side.
You need to debug the code and find a way to make it work in the way you need the business task to be completed. Probably if you could share the code of your page and describe the task in details we could take a look at it together.
If you expand this AcitvityMiniPage to the normal page (ActivityPageV2) the record is saved automatically. For this reason it is always a "change" for the system if I click on the "Save" button.