There is some cases of ambiguity with plus, but your example of « je voudrais plus de la soupe » seems a bit broken, it would sound strange and ill-formed in any context anyway. You can have either :

Je voudrais plus de soupe. (more of it, no ambiguity)

Je ne voudrais plus de soupe. (no more soup, no ambiguity)

(bold s where it would be pronounced if spoken out loud)

But yes, ambiguity can occur when the first part of the negation is dropped in casual forms of speech :

Je [ne] veux plus de soupe. >>> Je veux plus de soupe.

Here I switched tense to simple present to fit the casual context because "voudrais" sounds slightly more polite and formal, but the point is the meaning depends on how it’s pronounced, so in conclusion, yes, « Je veux plus de soupe. » has an ambiguous meaning in written form.

Regarding the part about un peu plus, I admit I never noticed this usage of the expression to avoid the ambiguity.

There is rarely an ambiguity in writing because a negative sentence contains the particle ne, whereas a positive sentence doesn’t.

Je veux plus [plys] de soupe.   (I want more soup, s is sounded.)
Je ne veux plus [ply] de soupe.   (I don’t want any more soup, s is silent.)

The particle ne can be omitted in colloquial speech, but this is not done in writing except to transcribe colloquial speech.

On the pronunciation of plus, see also When does one pronounce the 's' in plus?

“*Plus de la soupe” is not correct. The partitive article in French is a bit complicated. It is normally du/de la/de l’/des, but when the existence is negated, it is de instead. This is the case in “je ne veux plus de soupe”: since the soup is negated, the partitive article is de. In “je veux plus de soupe”, the word de is a preposition introducing a noun complement, and when it is used to indicate the composition or purpose of an object, there is no article after de. This is the case in particular when expressing a quantity, thus “plus [plys] de soupe” (more soup), “moins de soupe” (less soup), “un peu de soupe” (a little soup), “un bol de soupe” (a bowl of soup), “un litre de soupe” (a liter of soup), etc.

For more information about the partitive article, see also partitif, in particular Usage of “d'eau” vs “de l'eau” and Why don't “des”, “de la”, “du” always become “de” in negative sentences?

The main point here is this:

It’s important here to remember that:

ne plus vouloir quelque chose: to no longer want something
[silent s] is not

ne pas vouloir plus de quelque chose; to not want more of something. [s heard]

Following your reasoning:

1) C’est un plus de pouvoir parler une langue étrangère. [“positive”;noun]
2) Je ne sais plus quoi faire. [“negative”: adverbial]

Mean:

1) It’s a plus [noun] to be able to speak a foreign language.
2) I no longer know what to do. [ne…plus around the verb; an adverb for no longer]

To ask for more soup, plus de soupe functions like an adjective:
Je voudrais un peu plus de soupe. Plus de, moins de do not take an article:

plus de soupe, moins de soupe, plus d’argent, moins d’argent, they are adjectives used with the partitive articles for uncountable nouns. More soup, less money, etc.

Following your logic: “But if you wanted to ask for more soup, wouldn’t this be ambiguous in writing? For example “je voudrais plus de la soupe” could mean either you want more soup or you want no more, depending on whether you pronounce the s or not?”

Answer: Not exactly. It cannot mean no more. See the end of this comment.

Grammatically, it is: Je voudrais plus de soupe. Or: Je voudrais plus d’argent. I would like more soup or more money [adjectives]. However, to avoid this quasi-rude form, the French tend to say: Je voudrais encore un peu de soupe or un peu de soupe encore or even un petit peu plus de soupe. [adjectives]
It is also to avoid the confusion of ne plus vouloir de and ne vouloir plus de.

Grammatically, for: I do not want more soup, it would be: Je ne voudrais ou veux pas plus de soupe. However, bear in mind, the question will usually be: Est-ce que tu veux ou vous voudrez encore de la soupe? And the answer is usually: Non, merci, je n’en veux plus ou Je n’en voudrais plus. It is not ungrammatical to say: Je ne veux pas plus de soupe. But it is not usual.

Your question again:
Is there a difference between “plus de soupe” and “plus de la soupe”?
But it all depends on what question.

Je ne veux pas plus de soupe. I don’t want more soup. Usually: Je n’en veux plus.

Je ne veux plus de soupe. I no longer want soup.

Je ne veux plus de la soupe. I no longer want any of the soup [being served here].

[I hope I straightened that out for an English speaker.]

In polite French, the conditional tense is used rather than the regular present.