That sentences you wrote look identical. Did you mean to make a contrast?

You wrote the same sentence twice. Are you asking if it's ambiguous in some specific way? (All sentences can get different meanings in different contexts. That's what contexts do.)

The "meaning" of prepositions is often vague (notoriously hard to translate, for example). The default interpretation of those two sentences would be the same. However, "by" means "somewhere near", while "at" means roughly "at the end of a path leading to it".* So "by" might allow a wider range of positions, although by default it would be assumed to also be "right next to", like "at". There isn't any more substantial difference between the sentences, just some semantic flexibility in spatial prepositions.

[*Note: "at" does not necessarily imply motion of the object described, although in most circumstances at least indirectly it probably does. You can say "I am at my office", which doesn't directly refer to motion but does indirectly imply that I was somewhere else previously. With inanimate objects you might be able to come up with a sentence that doesn't imply motion, but even then it often might. "The question mark is at the end of the sentence" I suppose does not imply motion, except in the sense of the writer's hand placing it there at some point. Regardless, the meaning of "at" seems somewhat similar to the meaning of the verb "arrive" to me, since if you have not yet "arrived" somewhere, then you are not yet "at" that place. "By" just means "near, close to", so it is interpreted relatively.]

As I see it, "at" implies "in the same general place as", whereas "by" implies "in a place adjacent to". "At" may or may not mean "in", but "by" never does.

Compare "There is a coffee machine at [i.e. in] the library" with "There is a coffee machine by the library".

Ah, yes, you're absolutely right. But the "gate" example here didn't make me think of it that way. I don't imagine a gate as something you can be "in" (although that's possible in some sense), and "by" could mean "by" either of the posts/walls making up the gate. But the library example is a good one.

Actually now that does bring up another use of the phrase "at the gate" (specialized or idiomatic), in reference to a competitor at the starting point and ready to begin a race. But I don't assume that's what the original question was asking about.

On the question of motion, I think "at the beginning" is an example where no motion is implied.

That's one example of a common pattern of spatial expressions being used temporally. 'Time is motion' as the cognitive linguists would say. So it just depends on which dimension we're looking at-- here it's displacement in time, not displacement in space.

No, but it implies future (temporal) motion.

The equivalence with "arrive" is not exact (especially temporally), but rather that it has a sense of being "at" rather than "near". That is, at the extreme of a path, can't get closer in any relevant sense.