Read the whole article. The answer is already there.
There are two forms of “any”: a weak and a strong. You have posted the definition for one (the “weak form”), but are confused when you see the other (the “strong form”). But if you continue to read the rest of the articles you linked, you will find the answer is there.
Your examples illustrate the two forms of “any”
Your two samples are an example of the strong and weak forms respectively:
Your interpretation, “…the first one implies the asker expects exactly one patient, the second one implies he expects more than one patients.” is right for the second case, but wrong for the first.
In the first, strong, sentence the speaker is asking about the existence of any patient, but they expect that there isn’t one, so they’re challenging the listener for a response. (“challenging” doesn’t mean they are being aggressive, just that by asking the question, they are making a claim).
In the second sentence, the use of the weak form, suggests that the speaker expects that there may be such patients, and is asking for confirmation. Here, the questioner makes no claims, and is just requesting information.
You should use relative clauses in questions like these
While it is possible to figure out your meaning here, the wording you have chosen is not natural. Instead of “not responding to...”, native speakers would use a defining relative clause here, as follows:
…note that “who” is the same for singular and plural subjects, but the verb “to respond” still has to change to match the subject.
“strong” and “weak” are spoken differently
The strong and weak forms are much easier to spot in speech:
- strong: Is there any patient not responding...?
- weak: Are there any patients not responding...?
Note that the “strong form” any is stressed, while the weak one has the word “any” unstressed.
This “rule” is general advice about the most common use of “any” to stop you making a common learner’s error
The weak form is the more common in writing and speech. The article is trying to steer you away from making clearly inocorrect statements like this:
- wrong: Is there any chip in that packet?
- wrong: I do not have any holiday left this year.
- wrong: Is there any potato in the pantry?
- wrong: We cannot sell any knife to you.
These are questions, or negative statements about quantity or number. In this kind of sentence, where there’s no emphasis, when you use “any X”, then the general rule is that X should be plural if it is a countable noun.
This is a good rule to learn first, but it is not by any means the only rule for the use of the word “any”, and your articles do go into several exceptions to this advice: just scroll down.
…some nouns look singular, but aren’t
Also, consider that there are nouns that are both countable and uncountable depending on the meaning:
- correct: He doesn’t have any heart.
Why not “hearts?”. Well, it could be, if we were talking about a butcher, but here, “heart” is the uncountable noun meaning “compassion or kindness”, not the organ that pumps blood.
…but this does not apply to every kind of sentence with the word “any”
But this rule applies only to questions, or negative statements using what’s called the “weak” form of “any” - where you are not trying to claim anything by asking the question. The other form, the “strong form” is used to say you are not talking about the quantity, but about whether or not the thing exists. When you use the strong form, you are also making a statement about what you think the answer is (usually it’s “no”). Here are some examples:
- correct: I don’t not know if he had any home in those days.
- correct: Any problem that might occur would be minor.
- correct: Is there any among you who would help me?
#3 is a different use of “any”, with the meaning of “any [person]”. It’s a little archaic, but because it is in English translations of the Christian Bible, and the works of Shakespeare, it still gets used sometimes by speechmakers (or dialogue for fantasy films...)
Other uses of “any”
These sentences are the “strong” form of any, but are not questions or negative statements, so there’s no special meaning if you use the singular or plural.
- correct: You can have any colour you want.
- correct: You can have any colours you want.
- correct: “Any port in a storm”
Note #1 implies that the thing can be a single colour of your choice, #2 implies that one or more colours could be used.
#3 is a saying. The meaning is that when your situation is bad, you do not have the luxury of choosing who helps you: when the sea is stormy, even an unsafe port is a better place for your ship. This is a similar meaning to “beggars can’t be choosers” or the very idiomatic “needs must when the Devil drives”)
The strong form of “any” in questions and statements
Even in questions and negative statements, native speakers do indeed use singular nouns with “any”. Doing so is a sign that they’re using the strong form. Generally this is indicates:
- rhetorical questions
- very emphatic negative statements
- Is there any waiter working in this restaurant tonight?
- Is there any reason why I shouldn’t fire you right now?
- There isn’t any tire for driving on ice (i.e., “such a thing does not exist”)
- No, there isn’t any ticket (i.e., “you asked me for tickets to one show, there are in fact no tickets at all, regardless of the show”)
Remember that the strong form makes a claim as well as asking a question - the questions are basically re-phrasings of the emphatic “There is no X” (where “is” is stressed). Also, note that this kind of strong emphatic or rhetorical use can seem impolite.