One picture holds all three uses: two sides on opposite sides of a border, each handing over what is theirs and taking back the other side's version of the same thing. Usually what crosses is money — my dollars for your euros — and the words then cover both the trading and the money that comes in from it. Once, what crosses is people: my school year in your country, yours in mine. The little word a is what tells you which crossing is meant, because the money uses take no article and the student one does.
Two branches
錢過境：換錢這件事，和換進來的錢
最常見的一面：被交換的是貨幣，而同一組字同時指這個動作和它的成果。
noun · B2
The buying and selling of one country's money for another country's money, and the whole business built around it. It is uncountable, and it very often sits in front of another noun: foreign exchange market, foreign exchange rate, foreign exchange desk.
外匯（買賣外幣這件事）
人過境：交換學生
同樣是兩邊對等互換，只是被交換的不是錢，而是人和他們的位置。這是理解上的連結，不是歷史來源的說明。
noun · B2
An arrangement in which a student goes to live and study in another country, and a student from there does the same in return, usually for a set period and often in order to learn the language. It is countable.
（學生）交換