Your first interpretation is correct.
From "What is an Exchange Student?" by Kathleen Crislip:
An exchange student is a high school- or college-aged student who travels abroad to live in a new country as part of an exchange program. While they're in this program, they'll be staying with a host family and attending classes at the local school, all while immersing themselves in a brand new culture, potentially learning a new language, and exploring the world from a different viewpoint. It's a fantastic opportunity and one I recommend all students grab with both hands.
Note that, despite the name, an exchange trip (or exchange program) does not necessarily mean that one student goes somewhere and another student from that other place is sent to the first place in return (or exchange). It can mean that, but exchange trips can also happen where only one student goes in one direction. Although it's assumed there will be reciprocity for other students to come the other way at some point too—even if not necessarily at the same time.