There actually is a difference in English (even just meaning-wise, ignoring grammar). "The small dogs are drinking" means that if you look at all the small dogs in the room, they're all drinking, but big dogs may or may not be drinking. Meanwhile, "the drinking dogs are small" means that if you take all the dogs that are drinking, they must all be small. So in the scenario where all the small dogs are drinking but some big dogs are drinking, the former sentence makes sense while the latter sentence doesn't. Sorry if I overexplained it, I just wanted to be thorough.
I concur that the sentence taught as most correct is jarring. The alternative, accepted and I would say more correct, translation is:- The dogs drinking are small. I am also doubting the ~ing as gerunds are in a few chapters time so there should be a wh~ word in the translation sentence. So all in all the deemed correct translation is poor.
They don't mean the same. In this sentence, the verb of the sentence is 작다. So, [something] is small. What is small? [some kind of] dogs are small. What kind of dogs are small? The ones that are drinking. So, the final outcome is "The dogs that are drinking are small" which is the same as "The drinking dogs are small". This sentence differs from "The small dogs are drinking" by the fact that the observer has discovered that the size of the dogs that are drinking is small as opposed to them discovering that the small dogs are doing the act of drinking.
The small dogs are drinking would be "작은 개들은 마셔요" or, to preserve the progressive tense, "작은 개들은 마시고 있어요".
The main predicate (verb/adjective) of the sentence goes at the end. "The small dogs are drinking" would be 작은 개들은 마셔요 because the main verb is "drinking" (마시다), so it should go at the end, and "small" (작은) is just a modifying word, so it goes before the subject "dogs" (개들).
The order in the sentence here, 마시는 개들은 작아요, tells you that "drinking" is a word modifying "dogs", and the main predicate is "small"—hence, "The drinking dogs are small" (or, "The dogs who are drinking are small").