In general, when leave ends, that's when mids have to be back. It may be different for different classes, or even varsity sports in season, especially at Reform in the late summer.
"Intersession" is a catch-all term used in different ways at different times, and the "inter" part can be misleading. It can describe a few days between end of leave and actual start of classes, a time for briefings, Commandant's Calls, class schedule or course switching and other administrative events. I think that is most likely the case for the dates you describe above.
There has been in the past, at the end of spring semester, a window of time between end of classes and the activities of Commissioning Week, called "intersessional leave," a period of time when mids may be allowed to take a leave break, depending on their schedule.
Your mid will be responsible for knowing when and where he or she has to be.