When I (re)schedule a past appointment of 60 minutes (for example), in the appointments section of the client, the system shows that all the minutes are left. How can I mark them as spent?

That's a default setting, since the system has no way of knowing in advance whether you spent all those appointment minutes with the client (or some of the minutes).

So, what you can do is to use the "edit duration". Click on the meatballs icon (three dots) select "Edit Duration". Enter the time interval spent with the client (in minutes) and save. 

If you've had a 60 minutes appointment and you've spent all the 60 minutes, that is the value you need to enter as duration in the timer settings. When you refresh the page after that, you will see that there are 0 minutes left out of the 60.