I was just about to ask this question and saw that you already asked it. I only have hypotheses.
When I first read the terminology for the progressive/continuous aspect in English, it was be + gerund. That struck me as strange, since it seems that when you say “The teacher is talking,” the word talking modifies the teacher as adjectives normally modify nouns, as in “The teacher is tall” and “The teacher talking right now is Mrs. Lewis.” So, the -ing word seems to be functioning as a present participle. Indeed, today it’s more popular to call it a present participle.
Why two grammatical terms?
Considering that the gerund and the present participle always have the same form in English, where did people get the idea to call it by two names? The obvious suspect is people trying to understand English grammar in terms of Latin grammar—or perhaps by analogy with Romance languages. Italian has a present progressive and distinct forms for the gerund and present participle. And indeed, in Italian you say L’insegnante sta parlando (with the gerund), not L’insegnante sta parlante (past participle). But in English, the two forms are the same, so maybe the very question of which term to use for the -ing word in a progressive tense is pointless hair-splitting, resulting only from trying to force-fit English grammar terminology to foreign grammatical distinctions. Better to just call it “the -ing word” and be done with it.
On the other hand, just because two words have the same form doesn't mean they serve the same grammatical function. You modify a present participle with an adverb (“loudly talking") but you modify a gerund with an adjective (“loud talking”). You modify a verb in a progressive tense with an adverb (“The teacher is talking loudly”), so that would seem to seal the case for having two grammatical terms and for saying that the -ing word in progressive tenses is a present participle.
More grammatical functions
But wait! After looking at how the gerund and present participle are used in Italian, it’s clear that there are still more, decidedly distinct grammatical functions here. Consider the following:
A rolling stone gathers no moss.
A grinding stone is very heavy.
Rolling refers to the time when the stone is failing to gather moss. This is what the gerund means in Italian outside of progressive tenses: something happening while the main verb is happening or in connection with the main verb. While the stone is rolling and because the stone is rolling, it gathers no moss.
Grinding is an adjective describing the kind of stone. This is what the present participle means in Italian: it really is a way of converting a verb into an adjective, not a way to predicate something of a noun (the essential function of verbs).
Notice that in the present progressive tense, in both English and Italian, the verb predicates some action of the subject right now. This seems like a reasonable argument for saying that the present progressive in English is formed from be + gerund, not be + present participle, though of course the argument for the latter is pretty strong, too.
More grammatical restrictions
And can you really not modify a gerund with an adverb? We say “Loud talking is annoying” but also “Talking loudly is annoying.” “Loudly talking is annoying” is strange but maybe grammatical, and “Talking loud is annoying” is decidedly ungrammatical. Yet talking is clearly a noun in every sentence, and therefore seemingly a gerund.
My car is running.
Interpreted as a gerund, that would mean that your car’s engine is on right now. Interpreted as a present participle, that would mean that you have a running car right now, that is, a functional car.
So, my current thinking (tonight) is that the progressive tenses are made with be + gerund after all, contra my initial disbelief. Only, this is a gerund functioning as a verb, not a noun.
Another test of this hypothesis:
The children are describing talking with you at Chuck E. Cheese.
The children are describing talking robots at Chuck E. Cheese.
In the first sentence, talking is a gerund. In the second, talking is a present participle. And so describing is a gerund in both: it's really part of the verb, saying what the children are doing, not an adjective modifying children.
On the other hand, the fact that the gerund also functions as a noun (the Romance gerunds can’t do that) and the weird quasi-rules about where you need to put adjectives and adverbs make me think that the -ing form of a verb is just the usual English grammatical mish-mash: it serves many different roles, by various, sometimes conflicting, analogies with a variety of familiar constructions that serve as precedents, sometimes simultaneously. An ideal terminology might have to sort out the roles independently of the syntax, and allow a single -ing to serve more than one role at once.
There are two roles like Romance-language gerunds (present progressive and “something happening in connection with something else that’s happening”), there’s a role as a noun that works almost the same as an infinitive (also called “gerund”), there’s a role as an adjective unrelated to the time of the main verb, and maybe there are more. In a sentence like:
I am imagining playing the piano.
you could say that playing is a noun, the object of imagining, or you could say that playing is a gerund asserting an action that’s connected with imagining, or you could say that playing is a subordinate gerund, or you could even say that imagining playing is a compound present progressive verb. Just as with the many interpretations of quantum mechanics that are consistent with all observations, the language doesn’t provide evidence to say that one of these theories is right and the others are wrong.
Please, someone explain to me why this is wrong, either in a comment or another answer!