Difference between an adverb modifying an NP consisting a single noun, and an adverb modifying a noun

On the face of it, your sentences 1 and 2 seem extensionally identical -- that is, each is true if and only if the other is true. So I don't know whether I can find an answer. But I'll discuss it.

Syntacticians think about sentence structure in a peculiar way -- a way that traditional grammarians usually do not think about it. Syntacticians take sentence structure as something that actually exists and can be discovered, while other people generally think of it as a mere taxonomy -- a convenient classification system for discussing grammar. So it is often difficult for me to explain to a person trained in traditional grammar what the difference between a noun and a noun phrase is. (Hereafter N and NP.)

To a traditional grammarian, it's straightforward. A NP has to have more than one word in it, because that's what "phrase" means. A syntactician who has experience writing phrase structure grammar doesn't look at it that way at all. The difference between NP and N has nothing to do with the original source of the "P" in the name "NP". If I want to ensure that an NP always has more than one word, I have to make sure that I always write rules like NP -> old men but that I never write a rule like NP -> men.

This is not easy, when I want to start generalizing the rules. If I write NP -> Det N, I have to make sure other rules expand both Det and N as at least one word each. If I want to allow for sentences that have one word subjects, I can't have just the rule S -> NP VP -- I have to add the rule S ->N VP.

After a while, you start to wonder what the point is. Do one word subjects display any grammatical behavior distinct from that of multi-word subjects, other than having just a single word? Well, no.

There is also a difficulty with two tempting generalizations about English coordination. Ordinarily, you can only coordinate things of the same grammatical category, but then what about "socks and old shoes"? If a single word is of a different category from a multi-word phrase, this example should not be allowed. Furthermore, ordinarily the coordination of two like categories is of the same category as each of the things coordinated. The coordination of two verb phrases is a verb phrase, e.g. But an exception would have to be "socks and shoes", if I distinguish between one-word and multi-word categories, because I coordinated two single words and got a multi-word category.

So that's a problem with the sentence 2 in your question. You've proposed a difference in grammatical behavior that depends on whether a constituent has more than one word in it. But so far as I know, English just doesn't work this way, though it's not impossible that it might, I suppose.