Jackson6612Let me rephrase it again, even indefinite article the is used with countable nouns or the nouns which are acting as counting nouns.
What's "indefinite article the"?
Jackson6612 So it means if some non-countable noun is acting as a countable noun then it has to be an uncountable because there are two main types of nouns: countable and uncountable.
Sure. Isn't it a tautology? Dictionaries describe "happiness" as an uncountable noun, but in specific contexts (like GG's example) it may be used as a countable noun, which I have called playing a "countable" role, or, to be more correct, the role of a countable noun. So, what the dictionaries say doesn't always 100% correspond to reality.
Jackson6612 You said, pour the water onto the tea. Does that mean though water is an uncountable noun but in that particular example it is working as a countable noun?
Actually I used that example to show you a usage of an uncountable noun as an uncountable noun. "The" "defines" "water". It indicates that it is not any water, but that very water that has just been boiled. Water is still uncountable here, although it is "restricted" (specific).
Jackson6612 You said, it denotes a specific instance of that general category. I would have written: ...it denotes a specific type/kind/sort of that general category at some particular time. Would that also mean the same thing?
Well, I think it is a matter of one's Weltanschauung, and personally I prefer "instance" because, as distinct from "type", which, as any unit of classification, is abstract by nature, is real to the same extent as that which it has been derived from:
A is a type/sort of B — here A is not as real as B,
A is an instance of B — A is not less real (material) than B
Jackson6612 In a good detergent, a is standing along a countable noun but detergent in itself is not a countable noun. Then, what does make it a countable noun in that particular sentence?
At last, you have asked a specific question. By "detergent" the speaker didnt' mean the "matter" or "substance" itself, but, rather, a type or sort (or brand) of it. "Tide" is one detergent, and "Surf" is another. In this sense, they are countable.
Jackson6612 Can you, please, provide me some example sentences in which there is no need to use articles with certain countable/uncountable nouns with whom articles would be used in other sentences.
Countable: [link]
As for uncountable, you have already seen them: "Water boils at 100 centigrades (at the "standard" atmospheric pressure)" — here "water" refers to the verty substance, and the sentence is true for all water in the world.