How are the Timer and the "Set duration" features to be used in the Appointments section of the client?

You can start the Timer at the beginning of the appointment with the client. When you are done, you click on the "Pause", then on the "Done" buttons. The Appointments page of the client will then refresh, thus recording the actual time spent with the client. Let's say the initially set time interval for the appointment was 45 minutes, but on Timer you spent 42 or 47 minutes. When you use the Timer, the appointment time interval will update the initial 45 minutes to the actual time spent.

The same way you can update the actual time spent with the "Set duration" link, if you recorded the time with some other device (like you hand watch) instead of the Console's Timer.