So actually both are quite fine when we mean some short distance from the river/lake?

They're not interchangeable. When you're at the river you have reached it, having traveled to get there. When you're by the river, you're next to it.

Doesn't "at" mean "near"? Consider, "We are standing at the door" and "We are standing by the door" - both mean "near", don't they?

@SovereignSun: You're near the right idea, but not at it yet.

Both the prepositions convey "near", "beside", "in a hand's reach", etc. Scrolling through a dozen examples with either of them I stumble upon most uncontemplated usages in various contexts.

Please feel free to persist in your ignorance.

When a thing or creature capable of locomotion is said to be at a place, it has reached the place and is now there. If you are at the door, you have come to the door.

You perfectly well know that it's not about ignorance but rather about misunderstanding.

Are we at an impasse or by an impasse?

Definitely "at an impasse" you can't be "by" one.

With all due respect to you I think in most cases "by" means "in the vicinity" or "somewhere near" while "at" means "in front of" or "near".

What's the difference between "Leave your guns at the door" and "Leave your guns by the door"?

Are you "in front of the river"? I'd be more inclined to say you're close if you had said "confronting the river".

Why don't you read what I wrote?? at is for locations (real or imagined locations): "leave your guns at the door". at any impasse***//***at does not mean near. at goes to location in space/geography. My answer provides examples. by means next to or alongside. If you are at a place, you are more than near it, you are "on top of it" so to speak. Near implies that you are not yet actually at the place.

@SovereignSun That's easy: "We were not at the school, but we were near it." What makes you think you can catch me out? You can't. However, you might learn something from me. "The parcels were not at the house, but they had been thrown in a field near it."

@Lambie That's different. I agree on this one. But if we are "at the gate" we mean in front of it and "near the gate" is somewhere close to it yet then "by the gate" also means somewhere close and probably very close. With lake at the lake will mean the whole area we consider to belong to the lake and not the inside of the lake as in at the house or at the school and by the lake will mean in the vicinity very close to the water so actually it is somewhere near it.

@SovereignSun I am not going to argue with you. You have something to learn from me. You obviously are not interested in baseline nuance. NO: "with lake at the lake will mean the whole area we consider to belong to the lake". That is 100% wrong. At the house, at the lake, at the school are exactly the same thing. And there is no "inside the lake"; there is only "in the lake" [swimming or drowning] or "on the lake [in a boat. etc.]. You still have not got this. SORRY.

What does it mean when you look at someone, or shoot an arrow at a target?

?? That is not the USAGE we were discussing.

@Lambie Now why do people say at the sea meaning the seaside and at the lake meaning the lakeside?

@SovereignSun People don't say at the sea. They say: We were at sea for three months. i.e. our location was in a boat on the sea (back to that AGAIN). Normally, at the seaside is the expression for one's location. He's at the lake in his lakeside house. But at the lake does not mean lakeside.

But @Lambie these are not unrelated, since at sometimes has the meaning "to be arrived at the goal or destination of a progress" (or at some point along the way). In such cases the locative meaning is not simply "situated there" but "having arrived or reached there".

"at sometimes has the meaning "to be arrived at the goal or destination of a progress" (or at some point along the way"?? NO, and that phrasing is not even English. Your examples were: to shoot an arrow at a target. to shoot an arrow at the place (location) where the target is. To look at: to look at the place where the person or thing is located.

@Lambie: you don't see the semantic connection between goal and target? progress and direction?

at is about location and direction.