1. He is in the river.

[For example, half of his body is in the water. Or he is swimming in the river. He may be enjoying scub-diving in the river.]

2. He is on the river.
[He is rowing a boat on the river. He is floating down on a log on the river. He may be sitting on a boat on the river.]

3. He is at the river.
[He is not in the river, but he is by the river....Is my explanation right? Would you check them, please?]

Yes, all your explanations are right.

=)

Your explanations for the three phrases are mostly correct. Here's a breakdown of each:

1. He is in the river: This means that the person is physically inside the water, either swimming, wading, or submerged to some extent.

2. He is on the river: This means that the person is on a floating object, such as a boat or a log, that is positioned on top of the water and moving with the current. They are not in the water but on something that is on the water.

3. He is at the river: You are correct in saying that this means the person is by the river or near its vicinity. It implies that the person is in close proximity to the river but not necessarily in or on it.

Overall, your explanations capture the general meaning of each phrase. Well done!