I am not sure the "Each student can meet 0 or 1 teacher" means what you are saying. The relationship of meetings for a teacher with a student is 1 to many makes sense, but only in specific contexts, e.g., a class/course. For a school, it does not (many teachers, and students that take many classes).
But if there is only one teacher and one course, the first solution is fine. That is, keep a list of meetings (though I might have some sort of timestamp/schedule id as a way of temporally separating the meetings). Then the link table Teacher_met_student can record all student teacher meeting for an organization.
For me, the missing piece is the Course, which would have 1 teacher (and IRL, this is too restrictive, many courses have multiple instructors), many students, and meetings would have teacher, course, student to aggregate the meetings in a particular context.