Szwagier, you and I seem to be cut from similar, if not the same cloth. I can see that you tend to look at things in much the same way that I do, although we seem to have reached different (tentative) conclusions in this particular case. I couldn't agree with you more about ad hoc 'explanations' and how often they seem to send us off in the wrong direction.
But...I'm going to disagree with your first point. My feeling is that it is extraordinarily
helpful to attach two different names to a single form, depending on what it is doing at the time it is used. The two different names implies that, despite appearances, one use is not the same as another use even if the form looks the same.
Take, for example (just to use something different than your pres. part. vs gerund example for the moment, although the principle is the same, I believe) a look at some common verb forms. I prefer to teach my students (but if someone can show me where I'm wrong, I'll be glad to listen) that every verb in English has five
different forms:
a basic form:
jump
a present simple form:
jump/jumps (the -s being an arbitrarily imposed anomaly probably caused by a historical feature of English which is no longer necessary....but there it is, and we use it)
a past simple form:
jumped
a past participle form:
jumped
and a present participle form:
jumping
Now, notice that the past simple form (for this particular verb, chosen because it illustrates my point) and the past participle form look like the same word. I take pains, in my classes, to point out that though they may
look the same, they are indeed different, because they are
not ever used in the same way. For instance, the past simple "
jumped" is always used alone, without an auxiliary. We cannot say, * "She can jumped over the fence." or * "I am jumped when the boss said 'jump'." However, the other
jumped, the past participle one, is
always used with an auxiliary, and therefore has a slightly different component meaning. We
do say, "I have jumped over that wall many times." But the
jumped in this sentence is not the same
jumped as appears in the two previous example sentences. And, by a similar kind of reasoning, the
jump in the basic form is not the same
jump as the one in present simple form...they are not used in the same way.
So I would say, given an -
ing form, one could not say whether it was a present participle or a gerund on the basis of that alone. One would have to observe it in use in order to make the distinction. I would not say that it is a matter context, however, except in the sense of syntactic context.
I did not really understand what you meant by 'nouns and adjectives are connected in a way that nouns and verbs are not'. Could you elaborate on that one a bit more, please.
So the basic point, here, it seems to me, is that 'present participle' and 'gerund' are two names for two
different things, despite their appearance of being the same.
If that is acceptable to you, then your point #2 disappears, I think. We cannot, it seems to me, parse "I am walking." in the same way as we do, "I am cold." Or, at least, I would not. To me, the first example is a subject and a verb...period. The second is a subject, a copula verb, and a predicate adjective.
Now, I suspect that you may take issue with me. Good. Maybe I can learn something.

Maybe some other gentle readers can too!
Larry Latham