in the lesson for attendre the two examples that should illustrate the differences are the same:

Bonjour Jules ! Actually they are slightly different, one is "attendre", the other one "s'attendre à". We used the same sentence to show the nuance between the two uses of "attendre" in French. I hope that's helpful!

I do understand the difference. The English translations are the problem, they are both the same. The first should "They are waiting for him to arrive, They're waiting for his arrival." That would clarify the difference. The latter examples in the lesson do make this distinction.