I see my understanding of the question was wrong. My apologies. I should have asked for clarification before writing my first answer. I will leave it standing for now because I think it is relevant, at least in part.
You want to double-check your understanding of "want somebody doing something" with regards to the word "want." You have questions about the following sentence because it uses the word differently than you think it should:
"I don't want anybody in the class talking during the test."
That sentence is perfectly correct. I know because my teachers said it--or similar statements--many a time. Admittedly, I never knew there were any rules about how to use the word "want"; one just uses it without thinking. But I am looking through your basic rules to see which fits best and why.
Want meaning ‘wish’ or ‘desire'
Given the sentence structure, it seems the most appropriate rule is "Want meaning ‘wish’ or ‘desire'." The following sentence is perhaps the closest fit to use for an example:
She said I could have her old bike, but I don’t want it.
The important part of this sentence is:
I don't want __.
What don't "I" want?
Answer: "it," meaning "her old bike."
The teacher in your sentence is also saying:
"I don't want _____________________."
What don't "I" want?
Answer: anyone in class talking during the test.
This differs with my first answer where I agreed that "want" means "need." In that answer, I focused on the actual meaning of the word. Now I am focusing on the grammar rules, which is what you have been doing all along.
Want meaning ‘need’
Here is the grammar rule re need:
Want meaning ‘need’
Two examples they provide are:
Your hair wants cutting. (needs to be cut)
That cupboard wants clearing out.
In my part of the world, we don't say it like that; we say, "It's time to cut your hair," or "Your hair needs a trim." Likewise with the cupboard:
"It's time to clear out that cupboard."
"That cupboard needs to be cleared out"
For the teacher to use "want" that way, the sentence would say something like this:
"This class wants no one talking during the test."
Saying it that way, the teacher would risk the students disagreeing, or finding it silly, in the same way as with the statement in my first answer, "I don't need anyone in the class talking during the test."
To do, or doing
Back to "want somebody doing something." Just now I reread that post. The answer says it doesn't really matter which one one uses: to do, or doing. I agree. What matters is the sentence one is saying, the context of the situation, and whom one is talking to. It also matters what it is one is asking for.
I think this applies to all cultures and all languages: A parent or teacher can use different language or terminology to demand or request something from children than children can use to ask for something from their teachers or parents. In other words, superiors have the power to withhold privileges and subordinates must be humble if they want their wishes granted. Thus, "I want" may have to be expressed somewhat differently depending who is saying it to whom. Maybe this is off-topic.
Back on topic. If it does not matter whether we say "to do" or "doing," is there a difference in meaning between (does it matter which I say):
"I don't want anybody in the class to talk during the test."
and
"I don't want anybody in the class talking during the test."
Both are grammatically correct. The first one might imply that students are not allowed to raise their hands to ask a question, while the second one might impart the message that students should not chatter among themselves. However, I can't be sure of this. A lot depends on the communication patterns of teacher and students, the general culture of that specific classroom, and how people in that geographical location generally speak. Language is not set in stone. I say give it a try and see if it works.
