Toki Pona: A Language with a Hundred Words (2015)

1. The Philippines / The United States / The Great Lakes / the poor

5.8.4(b), Fixed expressions containing the definite article.

"In such cases, it is largely arbitrary that the definite article is required rather than a bare noun."

(kaoD says 'I don't think most native speakers realize "poor" is an adjective here', but there's good reason for that -- it is not ordinarily possible to construct a the+adjective phrase.)

Examples: [17iii] I have (the) measles.; [17v] We caught the bus.

2. A means of production

This is not at all idiomatic (Marxist rhetoric invariably refers to the means of production), so this is a simple example of 5.6.2, The indefinite article a.

"The indefinite article a is the most basic indicator of indefiniteness for singular count nouns." (here, means)

Example: [9a] Bring me a ladder!

3. A number of hamsters.

5.3.3(a), Number-transparent quantificational nouns

"Both [number and couple] occur in singular form with an obligatory determiner (usually a, but others are possible []), and in addition number can occur in the plural, and take a limited range of adjectival modifiers"

"The definite article the does not occur with number in the sense we are concerned with here."

Examples: [58i] We found huge numbers of ants swarming all over the place.; [58v] An unusually large number of people have applied this year.

4. The number of hamsters.

5.6.1, The definite article the

"The definite article the is the most basic indicator of definiteness."

"Use of the definite article [] indicates that I expect you to be able to identify the referent -- the individual ladder, the set of ladders, the quantity of cement [or quantity of hamsters] that I am referring to."

In order for this phrase to be grammatical, you need to use number in its ordinary, non-quantificational sense, as in "That depends on the number of hamsters you've put in the cage".

Examples: [1] Bring me the ladder!; [6vii] They are interviewing the man who mows her lawn.

5. A shirt / a piece of furniture / a pair of pants

You appear to be talking about the difference between mass nouns (pants / bread / furniture) and count nouns (shirt). This is discussed in 5.2 ("Overview of noun classes and NP structure"), but has absolutely nothing to do with definiteness or choice of determiner. However, a cannot be used with mass nouns because a requires a count of one, and mass nouns cannot be counted.

6. A lion chased him. (a particular lion)

5.6.2, The indefinite article a

Exactly the same as in "a means of production".

7. A lion is a ferocious beast. (lions in general)

5.8.3(i), Generic interpretations

"The interpretation of singular indefinites in the same context, like a lion in [iia], is correspondingly "any lion that exists."

"The generic use of a singular definite like the lion is also possible in the context given in [iiia], but is an example of the restricted 'class' use of the definite article (see 5.8.4 below). If instead of lions we were talking about doctors, the definite singular would not generally be possible"

"In The lions are ferocious beasts, the plural definite the lions would also obligatorily be interpreted non-generically."

"With predicates that can only be applied to a set, a singular indefinite generic such as a lion is inadmissible"

Examples: [14i] Lions are ferocious beasts. ; [14ii] A lion is a ferocious beast.; [14iii] The lion is a ferocious beast.; [16c] This chapter describes the English noun phrase.

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I have no real point; this was mostly for fun.

However, I'm a little confused as to why "a means of production" / "the number of hamsters" / "a shirt" / "a lion chased him" are in your list, since they're typical examples of the most common use of the / a as described above.

There is some overlap between the poor and the class use of the definite article, but it's not a perfect match, because it isn't possible to use poor in the same sense without accompanying it with the.

In "the poor people", "poor" is an adjective. If you elide "people" and turn the phrase into "the poor", it has to be a noun (otherwise we'd have a noun phrase without a noun).

Though IMHO it's better analyzed as an adjective with an elided noun, but I'm afraid I lost that war ages ago :shrug:

But I personally prefer to analyze what makes sense, not what's written (considering language is language). I'd rather have a regular case with an ellipsis than "oh, it's just a fixed phrase" and a zillion other corner cases.

And how did that phrase come to be? Noun ellipsis in "the poor people" (or "the poor ones" to be more general).