When you make a payment through the Commerce administration interface; orders are not completed. I can see a Place Order button when viewing a basket order; that seems to do the trick. I have to do it in the code.
I know I have to react to the commerce_order.order.paid event but I don't know what to do there.
I found this article: How to programmatically update the order status? which suggests that I should "apply a transition". But the suggestion doesn't work.
I opened a new question on this because I specifically ask about "placing an order"; while this question was about simply changing the status of the College.