The meanings can overlap.
To "quote" is to repeat someone else's words, in a way that indicates that you are repeating someone else's words, as opposed to incorporating them into your own work without any explicit identification that they are copied. (This could be plagiarism or it could be an allusion, but that's a different issue.)
To "cite" is to reference some other work. You may be quoting it word-for-word or you may simply be referring to it. "Cite" can be used to mean "quote", but it would rarely be used if you gave a quote without a reference, and it can be used when you give a reference without a quote.
Examples: "As Winston Churchill said, 'I have nothing to offer but blood, toil, tears, and sweat.'" This is unquestionably a quote. It might be called a citation.
"According to Barclay's Famous Quotations, Winston Churchill once said, 'I have nothing to offer but blood, toil, tears, and sweat." This is both a quote and a citation.
"In his History of Britain, Charles Stover says that Churchill called on the British people to be willing to make sacrifices." This is a citation but not a quote. It is not a quote because we are not repeating any exact words.
And to be complete: If I wrote a novel, and at some point in the novel I write, "General Framnitz urged his soldiers to fight on. 'I have nothing to offer but blood, toil, tears, and the billions of dollars of loot that we will get when we take the city'", that would be an allusion and not a quote, because I am not making any clear reference to Mr Churchill, and because I'm adapting the quote for my own purposes.
A citation can be informal like my examples here, "As he said in ...", or it could be a more formal reference, like "See Stover, Charles. History of Britain. Fwacbar Publishing, 1964, pp 85-86" (The exact formatting varying depending on what style guide you're using, if any.)
Your assertions in the first part of your question are correct - we use the word 'brain' both to describe the organ inside our heads, and as a synonym for intelligence. So to quote your examples:
- Use your brain.
- He killed people and ate their brains.
Both refer to the physical organ, in singular and plural form, respectively.
In this case, 'brains' is a synonym for intelligence, so again you are correct.
Now,here is the problem that encountered : "We need to put our brain
to work to make sure it stays fit for long."
Should not this be brains and not brain ? Because we are talking about
our brain that is the brains of more than one people.
Ignoring the oddly worded ending to the sentence, I would argue that the sentence is not gramatically correct. In the first half of the sentence, the personal pronoun 'we' and possessive determiner 'our' imply that we are talking about more than one person, but in the second half of the sentence only one brain is being referred to, as evidenced by the singular 'it', in 'to make sure it stays fit for long.' So it would probably be best to pick a side...
Either
- "You need to put your brain to work to make sure it stays fit for a
long time."
or
- "We need to put our brains to work to make sure they stay fit for a long time."
would be good starting points.
Best Answer
Think of "in here" as a more specific version of "here". In many cases you can substitute "here" for "in here", without really losing any meaning.
However, "in here" is not a substitute for "here", as it means something is inside, or at least within, some fairly specific location. While it's possible in your second example, it sounds strange to use "in here" with something like an entire city, because it suggests the person or thing is either visible or can be located with a brief search.
Some examples of using "in here" to emphasis the idea of an enclosed area: