I don't get why you're basing this question on the sloppy terminology used by answers on this site (which is not a good source for rigorous linguistic analyses). As you said, you can just replace "noun" and "adjective" in those descriptions with "NP" and "AdjP": "a gerund is a verb-form used in a non-finite clause that functions as an NP, and a present participle is a verb-form used as the head of a phrase that functions as an AdjP".

The function of a phrase typically depends on the part of speech of the phrase's head, so I don't see why you it's obviously a "gross mistake" to suppose that the word "eating" in "The man [eating this cake]" might be grammatically distinct in some way from the word "eating" in "[Eating this cake] is easy." I haven't read the relevant parts of CGEL in a while, but doesn't it address some arguments for a distinction in the course of ultimately arguing against that distinction?

@sumelic I don't know what's wrong with citing another answer in EL&U (a site for serious language enthusiasts) especially when the question has as many as 39 upvotes and the answer 34 upvotes. And simply replacing 'noun'/'adjective' with 'NP'/'AdjP' won't make the gerund/participle distinction any logical, because the gerund/participle terminology is about a word level whereas the difference is about a phrase level.

The (supposed) difference may only be apparent at (or above) the phrase level, but the diagnostic tests for distinguishing (gerund-)participles from adjectives are mostly at the phrase level, aren't they? "Takes a direct object" or "is modified by very" are phrase-level tests for word-level categorizations.

@sumelic I'm not saying that 30-something upvotes guarantees the quality of an answer/question in this site, but in all likelihood an answer/question having a lot of upvotes has greater quality than an answer/question having a lot of downvotes. As for the downvote on my previous question(s), let me just say there are always bad apples in any site.

@sumelic As for the word/phrase-level thing, I beg to differ. When you say a verb can take a direct object whereas an adjective can't, or that an adjective can be modified by very whereas a verb can't, either of the tests is at a word level, not at a phrase level, because it's a word (verb or adjective) that's being tested.

You ask: Why does traditional grammar still use the 'gerund'/'present participle' dichotomy for the very same form 'V-ing'? Do you mean: Why do some modern-day grammarians still make this distinction? In which case, can you name (and ideally cite from) one or two who do.

@Shoe I'm not a professional grammarian, so I can't say with certainty if some modern-day professional grammarians still make this distinction, but I know there are many grammar books currently available to ESL/EFL students that still maintain the distinction. Besides, many answers in this site, including the one cited in the question, consistently use the distinction. I know most of those who write answers here are not grammarians per se, but still they contribute to the massive confusion among ESL/EFL students.

Well, there's a whole discssion to be had on the relevance of descriptive grammars such as the CGEL to English language learners. In my experience learners are not much interested in, or are even intimidated by, grammatical terminology beyond the basics. What they care about is how native speakers commonly express the thoughts they themselves wish to express and whether what they say or write is 'correct' (i.e. grammatical). In this particular case I think its best simply to refer to the ing-form of the verb.

@Shoe I've cited a couple of grammar books that make the distinction. I agree with you that the ing-form is better than 'gerund'/'present participle'. But my question is about the confusing distinction, not about the -ing form.

@Shoe Exactly. I think Swan is essentially answering your question of whether "some modern-day grammarians still make this distinction", and the answer is 'yes', although Swan himself may not be included in those grammarians.

Interesting topic! The logic in the top parts leaves too much open for me to answer the question directly. I say that as someone who would refer to every -ing verb form as a "present participle" if he could get away with it. Two issues: (1) to be persuasive, arguing that infinitives don't have similar terminology requires me to assume that all areas deserve an equal level of distinction. I don't believe that. (If I did, why is that not an argument for specifying infinitive terminology?) (2) Why can't eating in 2 stand alone as an adjective? Participles, used as adjectives, can post-modify.

In other words, it feels like there are at least three distinct questions here, and breaking this up might make it easier to answer. The phrasing is tricky (how to ask something that isn't only opinion-based), but here's what I see: (1) When the gerund clause is treated as a noun phrase, is the gerund used as a noun? (Or the same with the participle/adjective.) (2) What is the difference between these two infinitive uses? Do they have a name? (3) What benefit is there to distinguishing between gerunds and present participles? (Or something asking about its uses in teaching or scholarship.)

And in short (last comment) I agree there's something tricky about something being "used as a noun" (or noun phrase) but also functioning as a verb, or "used as an adjective" (or adjective phrase) but also functioning as a verb; I just think that there's a lot I have to have an opinion on before I can even say, "Hey, that should have the same term or a different term."

CGEL say not. They lump. But ACGEL (Quirk et al) give a more detailed analysis of the verb ... noun continuum, as they see it, with quite a few different intermediate stages. It depends on how broad-brush a treatment you think is best. People claiming they've got the only legitimate treatment (eg by not acknowledging that there are alternatives held by some) are arrogating. Though people reasonably arguing their treatment is best are fine.

"is there any other logic behind continuing to distinguish 'gerund' from 'present participle' especially at a word level"=not possible. Only in the context of a phrase, utterance or sentence.

"Brown's deftly painting his daughter was a sight to behold" (Quirk et al) occupies a tricky middle-ground. Those liking neat solutions pretend that rank-ordering distribution tests is neat.

Sometimes the same construction can be construed differently, as in the well-known example "Flying planes are/is dangerous."

The sentence below: "Therefore, it's a gross mistake to distinguish 'gerund' from 'present participle' simply because etc. misquotes your own quote from the beginning of your question. Which is: A gerund is a form of a verb used as a noun, whereas a participle is a form of verb used as an adjective (or as a verb in conjunction with an auxiliary verb).

I'd like an example where there is something other than: a gerund used as a noun, a participle used as an adjective OR as a verb with an auxiliary. Are there any at all?

@Lambie In I was sitting smoking, smoking is neither a gerund used as a noun, a participle used as an adjective, nor a verb with an auxiliary.

@JK2 That is your opinion: I was sitting hoping and wishing. Hoping and wishing take up most of my time. Not a gerund noun? Hmm.

@Lambie You asked for my opinion, and I gave you one. Now, you tell me what you think drinking is in I was sitting smoking. Do you think it's either a gerund used as a noun, or a participle used as an adjective, or as a verb with an auxiliary?

@JK2 I'm saying it can only be a gerund used as a noun since you can take it and use it as one. In other words, transferability shows it, im my opinion. That's all. And I said it politely, too.

@Lambie I don't know what you're talking about. Just because it can be a gerund used as a noun in one sentence doesn't mean it can only be a gerund used as a noun. The very reason traditional grammar has different names (gerund and participle) is because the same -ing form can play a different role in a different sentence. In Hoping and wishing take up most of my time, both Hoping and wishing are gerunds in traditional grammar. But that doesn't mean they can only be a gerund in any sentence. Clearly, they're not functioning as nouns in I was sitting hoping and wishing.

@JK@ You don't know what I am talking about? I never said what your second sentence says. Of course, they can function differently. But that doesn't change my mind in those two sentences.

@Lambie Simply saying "that doesn't change my mind" doesn't mean anything, unless and until you show me why you think they're gerunds in I was sitting hoping and wishing, which is unlikely. There's no way you can show me that they're functioning as nouns, and you know that. End of discussion.

@JK2 I said the reason is transferability. Hoping and wishing take up most of my time. Most of my time is taken up with hoping and wishing. I see no fundamental difference in terms of those being gerunds. But I am a bit tired of oneupmanship and aggression.