That description says 本人 is closer to "not someone else but that person" while 当人 is closer to "the person in question". But as a matter of fact, they are very often interchangeable. Actually, both 本人 and 当人 make sense in all of the four examples you gave.
Still, I see several differences:
- 当人 is relatively less common, and is preferred in formal or legal contexts.
- 当人 typically refers to someone who is involved in some concrete issue, problem, etc.
- Only 本人 can be used when it refers to "real/true X (as opposed to X's agent, guardian, look-alike, photograph, etc.)".
For example, 本人の話を聞く and 当人の話を聞く are roughly the same, but the latter sounds stiffer or businesslike, and may sound like this person is involved in some trouble. In casual contexts (e.g., choosing a birthday present for a child), 本人の話を聞く is normally preferred. But 本人 is not an informal word, so if you're in doubt, I think you can stick to 本人 most of the time.
If you see someone on the street who looks exactly like a celebrity, and want to ask if he is real, then you can ask (ご)本人ですか but not 当人ですか.
There is also a set phrase 当の本人, an emphatic version of 本人/当人 ("that person in question", "no one but he himself").